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Metamath Proof Explorer

Theorem camestres

Description: "Camestres", one of the syllogisms of Aristotelian logic. All ph is ps , and no ch is ps , therefore no ch is ph . In Aristotelian notation, AEE-2: PaM and SeM therefore SeP. (Contributed by David A. Wheeler, 28-Aug-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

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

Proof

Step	Hyp	Ref	Expression
1		camestres.maj	\|- A. x ( ph -> ps )
2		camestres.min	\|- A. x ( ch -> -. ps )
3		con3	\|- ( ( ph -> ps ) -> ( -. ps -> -. ph ) )
4	3	alimi	\|- ( A. x ( ph -> ps ) -> A. x ( -. ps -> -. ph ) )
5	1 4	ax-mp	\|- A. x ( -. ps -> -. ph )
6	5 2	celarent	\|- A. x ( ch -> -. ph )