Theorem bamalip

Description: "Bamalip", one of the syllogisms of Aristotelian logic. All ph is ps , all ps is ch , and ph exist, therefore some ch is ph . In Aristotelian notation, AAI-4: PaM and MaS therefore SiP. Very similar to barbari . (Contributed by David A. Wheeler, 28-Aug-2016) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

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

Proof

Step	Hyp	Ref	Expression
1		bamalip.maj	|- A. x ( ph -> ps )
2		bamalip.min	|- A. x ( ps -> ch )
3		bamalip.e	|- E. x ph
4	2 1 3	barbari	|- E. x ( ph /\ ch )
5		exancom	|- ( E. x ( ph /\ ch ) <-> E. x ( ch /\ ph ) )
6	4 5	mpbi	|- E. x ( ch /\ ph )