Theorem darii 2303

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	. . . . 5 |- A.x(ph -> ps)
3	2	spi 1753	. . . 4 |- (ph -> ps)
4	3	anim2i 552	. . 3 |- ((ch /\ ph) -> (ch /\ ps))
5	4	eximi 1576	. 2 |- (E.x(ch /\ ph) -> E.x(ch /\ ps))
6	1, 5	ax-mp 5	1 |- E.x(ch /\ ps)

Syntax hints:   -> wi 4   /\ wa 358  A.wal 1540  E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by:  ferio  2304