Theorem disamis 2314

Description: "Disamis", one of the syllogisms of Aristotelian logic. Some ph is ps, and all ph is ch, therefore some ch is ps. (In Aristotelian notation, IAI-3: MiP and MaS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)

Hypotheses

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

Assertion

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

Proof of Theorem disamis

Step	Hyp	Ref	Expression
1		disamis.maj	. 2 |- E.x(ph /\ ps)
2		disamis.min	. . . . 5 |- A.x(ph -> ch)
3	2	spi 1753	. . . 4 |- (ph -> ch)
4	3	anim1i 551	. . 3 |- ((ph /\ ps) -> (ch /\ ps))
5	4	eximi 1576	. 2 |- (E.x(ph /\ ps) -> 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:  bocardo  2316