Theorem disamis 2137

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 1536	. . 3 |- ( ph -> ch )
4	3	anim1i 340	. 2 |- ( ( ph /\ ps ) -> ( ch /\ ps ) )
5	1, 4	eximii 1602	1 |- E. x ( ch /\ ps )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1351   E.wex 1492

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bocardo  2139