Theorem darii 2565

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 2054	. . 3 |- ( ph -> ps )
4	3	anim2i 593	. 2 |- ( ( ch /\ ph ) -> ( ch /\ ps ) )
5	1, 4	eximii 1764	1 |- E. x ( ch /\ ps )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  ferio  2566