Theorem darii

Description: "Darii", one of the syllogisms of Aristotelian logic. All ph is ps , and some ch is ph , therefore some ch is ps . In Aristotelian notation, AII-1: MaP and SiM therefore SiP. For example, given "All rabbits have fur" and "Some pets are rabbits", therefore "Some pets have fur". Example from https://en.wikipedia.org/wiki/Syllogism . See dariiALT for a shorter proof requiring more axioms. (Contributed by David A. Wheeler, 24-Aug-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

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

Proof

Step	Hyp	Ref	Expression
1		darii.maj	|- A. x ( ph -> ps )
2		darii.min	|- E. x ( ch /\ ph )
3		id	|- ( ( ph -> ps ) -> ( ph -> ps ) )
4	3	anim2d	|- ( ( ph -> ps ) -> ( ( ch /\ ph ) -> ( ch /\ ps ) ) )
5	4	alimi	|- ( A. x ( ph -> ps ) -> A. x ( ( ch /\ ph ) -> ( ch /\ ps ) ) )
6	1 5	ax-mp	|- A. x ( ( ch /\ ph ) -> ( ch /\ ps ) )
7		exim	|- ( A. x ( ( ch /\ ph ) -> ( ch /\ ps ) ) -> ( E. x ( ch /\ ph ) -> E. x ( ch /\ ps ) ) )
8	6 2 7	mp2	|- E. x ( ch /\ ps )