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Metamath Proof Explorer

Theorem calemes

Description: "Calemes", one of the syllogisms of Aristotelian logic. All ph is ps , and no ps is ch , therefore no ch is ph . In Aristotelian notation, AEE-4: PaM and MeS therefore SeP. (Contributed by David A. Wheeler, 28-Aug-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

	Ref	Expression
Hypotheses	calemes.maj	\|- A. x ( ph -> ps )
	calemes.min	\|- A. x ( ps -> -. ch )
Assertion	calemes	\|- A. x ( ch -> -. ph )

Proof

Step	Hyp	Ref	Expression
1		calemes.maj	\|- A. x ( ph -> ps )
2		calemes.min	\|- A. x ( ps -> -. ch )
3		con2	\|- ( ( ps -> -. ch ) -> ( ch -> -. ps ) )
4	3	alimi	\|- ( A. x ( ps -> -. ch ) -> A. x ( ch -> -. ps ) )
5	2 4	ax-mp	\|- A. x ( ch -> -. ps )
6	1 5	camestres	\|- A. x ( ch -> -. ph )