Theorem darii 2145

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. (Contributed by David A. Wheeler, 24-Aug-2016.)

Hypotheses

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

Assertion

Ref	Expression
darii	|- E. x ( ch /\ ps )

Proof of Theorem darii

Step	Hyp	Ref	Expression
1		darii.min	. 2 |- E. x ( ch /\ ph )
2		darii.maj	. . . 4 |- A. x ( ph -> ps )
3	2	spi 1550	. . 3 |- ( ph -> ps )
4	3	anim2i 342	. 2 |- ( ( ch /\ ph ) -> ( ch /\ ps ) )
5	1, 4	eximii 1616	1 |- E. x ( ch /\ ps )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1362   E.wex 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  ferio  2146