Theorem dimatis 2171

Description: "Dimatis", one of the syllogisms of Aristotelian logic. Some ph is ps, and all ps is ch, therefore some ch is ph. (In Aristotelian notation, IAI-4: PiM and MaS therefore SiP.) For example, "Some pets are rabbits.", "All rabbits have fur", therefore "Some fur bearing animals are pets". Like darii 2154 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.)

Hypotheses

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

Assertion

Ref	Expression
dimatis	|- E. x ( ch /\ ph )

Proof of Theorem dimatis

Step	Hyp	Ref	Expression
1		dimatis.maj	. 2 |- E. x ( ph /\ ps )
2		dimatis.min	. . . . 5 |- A. x ( ps -> ch )
3	2	spi 1559	. . . 4 |- ( ps -> ch )
4	3	adantl 277	. . 3 |- ( ( ph /\ ps ) -> ch )
5		simpl 109	. . 3 |- ( ( ph /\ ps ) -> ph )
6	4, 5	jca 306	. 2 |- ( ( ph /\ ps ) -> ( ch /\ ph ) )
7	1, 6	eximii 1625	1 |- E. x ( ch /\ ph )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1371   E.wex 1515

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-ial 1557

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)