AxmPropNegIntro (1.10, Axiom): A10

AxmPropNegIntro (1.10) — A10

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

Num Entity Database Record (Vertical Field Table)

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`