Theorem cesare

Description: "Cesare", one of the syllogisms of Aristotelian logic. No ph is ps , and all ch is ps , therefore no ch is ph . In Aristotelian notation, EAE-2: PeM and SaM therefore SeP. Related to celarent . (Contributed by David A. Wheeler, 27-Aug-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

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

Proof

Step	Hyp	Ref	Expression
1		cesare.maj	|- A. x ( ph -> -. ps )
2		cesare.min	|- A. x ( ch -> ps )
3		con2	|- ( ( 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 )