Theorem cesaro

Description: "Cesaro", one of the syllogisms of Aristotelian logic. No ph is ps , all ch is ps , and ch exist, therefore some ch is not ph . In Aristotelian notation, EAO-2: PeM and SaM therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

	Ref	Expression
Hypotheses	cesaro.maj	|- A. x ( ph -> -. ps )
	cesaro.min	|- A. x ( ch -> ps )
	cesaro.e	|- E. x ch
Assertion	cesaro	|- E. x ( ch /\ -. ph )

Proof

Step	Hyp	Ref	Expression
1		cesaro.maj	|- A. x ( ph -> -. ps )
2		cesaro.min	|- A. x ( ch -> ps )
3		cesaro.e	|- E. x ch
4	1 2	cesare	|- A. x ( ch -> -. ph )
5	3 4	barbarilem	|- E. x ( ch /\ -. ph )