Obelisk Entity Database — 1. Propositional Logic (1.01–1.14)

Classical propositional calculus axiom schemas (A01–A13) and Modus Ponens. (Generated directly from newchecker.ML via html_generator.ML).

Formula Display:

Proof Trees:

1. Propositional Logic (1.01–1.14)

Classical propositional calculus axiom schemas (A01–A13) and Modus Ponens.

Num Entity Database Record (Vertical Field Table)

Title	A01 `#AxmPropWeakening` `#label-a01`
Classification	Axiom
Formula	0(⟦𝛂⟧→(⟦𝛃⟧→⟦𝛂⟧)) 0\(\href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`PA1_prf`
Scheme of Types	0`(⟦[𝛂@Fm.Expression]⟧→(⟦[𝛃@Fm.Expression]⟧→⟦[𝛂@Fm.Expression]⟧))`
Judgements	1[(𝛂 :Fm ), (𝛃 :Fm )] 1\(\href{../symbols/image-op.html}{[} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/comma-delim.html}{,} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/image-op.html}{]} \)
Verification	Postulated (Kernel Axiom) · Standalone Entry (`AxmPropWeakening`) → · (1.01) · `#AxmPropWeakening`

Title	A02 `#AxmPropFrege` `#label-a02`
Classification	Axiom
Formula	0((⟦𝛂⟧→(⟦𝛃⟧→⟦𝛄⟧))→((⟦𝛂⟧→⟦𝛃⟧)→(⟦𝛂⟧→⟦𝛄⟧))) 0\(\href{../symbols/delimiters-brackets.html}{(} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\gamma}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\gamma}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`PA2_prf`
Scheme of Types	0`((⟦[𝛂@Fm.Expression]⟧→(⟦[𝛃@Fm.Expression]⟧→⟦[𝛄@Fm.Expression]⟧))→((⟦[𝛂@Fm.Expression]⟧→⟦[𝛃@Fm.Expression]⟧)→(⟦[𝛂@Fm.Expression]⟧→⟦[𝛄@Fm.Expression]⟧)))`
Judgements	1[(𝛂 :Fm ), (𝛃 :Fm ), (𝛄 :Fm )] 1\(\href{../symbols/image-op.html}{[} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/comma-delim.html}{,} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/comma-delim.html}{,} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\gamma}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/image-op.html}{]} \)
Verification	Postulated (Kernel Axiom) · Standalone Entry (`AxmPropFrege`) → · (1.02) · `#AxmPropFrege`

Title	A03 `#AxmPropConjLeft` `#label-a03`
Classification	Axiom
Formula	0((⟦𝛂⟧∧⟦𝛃⟧)→⟦𝛂⟧) 0\(\href{../symbols/delimiters-brackets.html}{(} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/and-conn.html}{\land} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`PA3_prf`
Scheme of Types	0`((⟦[𝛂@Fm.Expression]⟧∧⟦[𝛃@Fm.Expression]⟧)→⟦[𝛂@Fm.Expression]⟧)`
Judgements	1[(𝛂 :Fm ), (𝛃 :Fm )] 1\(\href{../symbols/image-op.html}{[} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/comma-delim.html}{,} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/image-op.html}{]} \)
Verification	Postulated (Kernel Axiom) · Standalone Entry (`AxmPropConjLeft`) → · (1.03) · `#AxmPropConjLeft`

Title	A04 `#AxmPropConjRight` `#label-a04`
Classification	Axiom
Formula	0((⟦𝛂⟧∧⟦𝛃⟧)→⟦𝛃⟧) 0\(\href{../symbols/delimiters-brackets.html}{(} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/and-conn.html}{\land} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`PA4_prf`
Scheme of Types	0`((⟦[𝛂@Fm.Expression]⟧∧⟦[𝛃@Fm.Expression]⟧)→⟦[𝛃@Fm.Expression]⟧)`
Judgements	1[(𝛂 :Fm ), (𝛃 :Fm )] 1\(\href{../symbols/image-op.html}{[} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/comma-delim.html}{,} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/image-op.html}{]} \)
Verification	Postulated (Kernel Axiom) · Standalone Entry (`AxmPropConjRight`) → · (1.04) · `#AxmPropConjRight`

Title	A05 `#AxmPropConjIntro` `#label-a05`
Classification	Axiom
Formula	0(⟦𝛂⟧→(⟦𝛃⟧→(⟦𝛂⟧∧⟦𝛃⟧))) 0\(\href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/and-conn.html}{\land} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`PA5_prf`
Scheme of Types	0`(⟦[𝛂@Fm.Expression]⟧→(⟦[𝛃@Fm.Expression]⟧→(⟦[𝛂@Fm.Expression]⟧∧⟦[𝛃@Fm.Expression]⟧)))`
Judgements	1[(𝛂 :Fm ), (𝛃 :Fm )] 1\(\href{../symbols/image-op.html}{[} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/comma-delim.html}{,} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/image-op.html}{]} \)
Verification	Postulated (Kernel Axiom) · Standalone Entry (`AxmPropConjIntro`) → · (1.05) · `#AxmPropConjIntro`

Title	A06 `#AxmPropDisjLeft` `#label-a06`
Classification	Axiom
Formula	0(⟦𝛂⟧→(⟦𝛂⟧∨⟦𝛃⟧)) 0\(\href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/or-conn.html}{\lor} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`PA6_prf`
Scheme of Types	0`(⟦[𝛂@Fm.Expression]⟧→(⟦[𝛂@Fm.Expression]⟧∨⟦[𝛃@Fm.Expression]⟧))`
Judgements	1[(𝛂 :Fm ), (𝛃 :Fm )] 1\(\href{../symbols/image-op.html}{[} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/comma-delim.html}{,} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/image-op.html}{]} \)
Verification	Postulated (Kernel Axiom) · Standalone Entry (`AxmPropDisjLeft`) → · (1.06) · `#AxmPropDisjLeft`

Title	A07 `#AxmPropDisjRight` `#label-a07`
Classification	Axiom
Formula	0(⟦𝛃⟧→(⟦𝛂⟧∨⟦𝛃⟧)) 0\(\href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/or-conn.html}{\lor} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`PA7_prf`
Scheme of Types	0`(⟦[𝛃@Fm.Expression]⟧→(⟦[𝛂@Fm.Expression]⟧∨⟦[𝛃@Fm.Expression]⟧))`
Judgements	1[(𝛂 :Fm ), (𝛃 :Fm )] 1\(\href{../symbols/image-op.html}{[} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/comma-delim.html}{,} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/image-op.html}{]} \)
Verification	Postulated (Kernel Axiom) · Standalone Entry (`AxmPropDisjRight`) → · (1.07) · `#AxmPropDisjRight`

Title	A08 `#AxmPropDisjElim` `#label-a08`
Classification	Axiom
Formula	0((⟦𝛂⟧→⟦𝛄⟧)→((⟦𝛃⟧→⟦𝛄⟧)→((⟦𝛂⟧∨⟦𝛃⟧)→⟦𝛄⟧))) 0\(\href{../symbols/delimiters-brackets.html}{(} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\gamma}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\gamma}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/or-conn.html}{\lor} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\gamma}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`PA8_prf`
Scheme of Types	0`((⟦[𝛂@Fm.Expression]⟧→⟦[𝛄@Fm.Expression]⟧)→((⟦[𝛃@Fm.Expression]⟧→⟦[𝛄@Fm.Expression]⟧)→((⟦[𝛂@Fm.Expression]⟧∨⟦[𝛃@Fm.Expression]⟧)→⟦[𝛄@Fm.Expression]⟧)))`
Judgements	1[(𝛂 :Fm ), (𝛃 :Fm ), (𝛄 :Fm )] 1\(\href{../symbols/image-op.html}{[} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/comma-delim.html}{,} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/comma-delim.html}{,} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\gamma}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/image-op.html}{]} \)
Verification	Postulated (Kernel Axiom) · Standalone Entry (`AxmPropDisjElim`) → · (1.08) · `#AxmPropDisjElim`

Title	A09 `#AxmPropNegElim` `#label-a09`
Classification	Axiom
Formula	0((¬⟦𝛂⟧)→(⟦𝛂⟧→⊥)) 0\(\href{../symbols/delimiters-brackets.html}{(} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/neg-conn.html}{\neg} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/bot-const.html}{\bot} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`PA9_prf`
Scheme of Types	0`((¬⟦[𝛂@Fm.Expression]⟧)→(⟦[𝛂@Fm.Expression]⟧→⊥))`
Judgements	1[(𝛂 :Fm )] 1\(\href{../symbols/image-op.html}{[} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/image-op.html}{]} \)
Kernel Note	Negation Elimination (Contradiction to Bottom): (¬⟦𝛂⟧) → (⟦𝛂⟧ → ⊥).
Verification	Postulated (Kernel Axiom) · Standalone Entry (`AxmPropNegElim`) → · (1.09) · `#AxmPropNegElim`

Title	A10 `#AxmPropNegIntro` `#label-a10`
Classification	Axiom
Formula	0((⟦𝛂⟧→⟦𝛃⟧)→((⟦𝛂⟧→¬⟦𝛃⟧)→¬⟦𝛂⟧)) 0\(\href{../symbols/delimiters-brackets.html}{(} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/neg-conn.html}{\neg} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/neg-conn.html}{\neg} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`PA10_prf`
Scheme of Types	0`((⟦[𝛂@Fm.Expression]⟧→⟦[𝛃@Fm.Expression]⟧)→((⟦[𝛂@Fm.Expression]⟧→¬⟦[𝛃@Fm.Expression]⟧)→¬⟦[𝛂@Fm.Expression]⟧))`
Judgements	1[(𝛂 :Fm ), (𝛃 :Fm )] 1\(\href{../symbols/image-op.html}{[} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/comma-delim.html}{,} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/image-op.html}{]} \)
Kernel Note	Negation Introduction (Reductio Ad Absurdum): (⟦𝛂⟧ → ⟦𝛃⟧) → ((⟦𝛂⟧ → ¬⟦𝛃⟧) → ¬⟦𝛂⟧).
Verification	Postulated (Kernel Axiom) · Standalone Entry (`AxmPropNegIntro`) → · (1.10) · `#AxmPropNegIntro`

Title	A11 `#AxmPropExFalso` `#label-a11`
Classification	Axiom
Formula	0(⊥→⟦𝛂⟧) 0\(\href{../symbols/delimiters-brackets.html}{(} \href{../symbols/bot-const.html}{\bot} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`PA11_prf`
Scheme of Types	0`(⊥→⟦[𝛂@Fm.Expression]⟧)`
Judgements	1[(𝛂 :Fm )] 1\(\href{../symbols/image-op.html}{[} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/image-op.html}{]} \)
Verification	Postulated (Kernel Axiom) · Standalone Entry (`AxmPropExFalso`) → · (1.11) · `#AxmPropExFalso`

Title	A12 `#AxmPropExcludedMiddle` `#label-a12`
Classification	Axiom
Formula	0(⟦𝛂⟧∨¬⟦𝛂⟧) 0\(\href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/or-conn.html}{\lor} \href{../symbols/neg-conn.html}{\neg} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`PA12_prf`
Scheme of Types	0`(⟦[𝛂@Fm.Expression]⟧∨¬⟦[𝛂@Fm.Expression]⟧)`
Judgements	1[(𝛂 :Fm )] 1\(\href{../symbols/image-op.html}{[} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/image-op.html}{]} \)
Verification	Postulated (Kernel Axiom) · Standalone Entry (`AxmPropExcludedMiddle`) → · (1.12) · `#AxmPropExcludedMiddle`

Title	A13 `#AxmPropVerum` `#label-a13`
Classification	Axiom
Formula	0⊤ 0\(\href{../symbols/top-const.html}{\top} \)
SML Kernel AST	`PA13_prf`
Scheme of Types	0`⊤`
Kernel Note	Verum / Truth Introduction Axiom (⊢ ⊤).
Verification	Postulated (Kernel Axiom) · Standalone Entry (`AxmPropVerum`) → · (1.13) · `#AxmPropVerum`

Title	Modus Ponens `#RulModusPonens` `#label-modus-ponens`
Classification	Inference Rule
Formula	0(⟦𝛂⟧, (⟦𝛂⟧→⟦𝛃⟧) ⊢ ⟦𝛃⟧) 0\(\href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/comma-delim.html}{,} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/impl-conn.html}{\rightarrow} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/turnstile-derivability.html}{\vdash} \href{../symbols/antiquotation.html}{[\![ } \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/antiquotation.html}{ ]\!]} \href{../symbols/delimiters-brackets.html}{)} \)
SML Kernel AST	`PMP_prf`
Scheme of Types	0`(⟦[𝛂@Fm.Expression]⟧, (⟦[𝛂@Fm.Expression]⟧→⟦[𝛃@Fm.Expression]⟧) ⊢ ⟦[𝛃@Fm.Expression]⟧)`
Judgements	1[(𝛂:Fm ), (𝛃:Fm )] 1\(\href{../symbols/image-op.html}{[} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\alpha}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/comma-delim.html}{,} \href{../symbols/delimiters-brackets.html}{(} \href{../symbols/formula-metavariables.html}{\boldsymbol{\beta}} \href{../symbols/typing-colon.html}{:} \href{../symbols/fm-sort.html}{\mathsf{Fm}} \href{../symbols/delimiters-brackets.html}{)} \href{../symbols/image-op.html}{]} \)
Side Condition	1⊤ 1\(\href{../symbols/top-const.html}{\top} \)
Verification	Primitive / Derived Rule (Kernel) · Standalone Entry (`RulModusPonens`) → · (1.14) · `#RulModusPonens`