Theorem disamis 2108

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	. . . 4 |- A. x ( ph -> ch )
3	2	spi 1516	. . 3 |- ( ph -> ch )
4	3	anim1i 338	. 2 |- ( ( ph /\ ps ) -> ( ch /\ ps ) )
5	1, 4	eximii 1581	1 |- E. x ( ch /\ ps )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1329   E.wex 1468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  bocardo  2110