Theorem datisi 2051

Description: "Datisi", one of the syllogisms of Aristotelian logic. All ph is ps, and some ph is ch, therefore some ch is ps. (In Aristotelian notation, AII-3: MaP and MiS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)

Hypotheses

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

Assertion

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

Proof of Theorem datisi

Step	Hyp	Ref	Expression
1		datisi.min	. 2 |- E. x ( ph /\ ch )
2		simpr 108	. . 3 |- ( ( ph /\ ch ) -> ch )
3		datisi.maj	. . . . 5 |- A. x ( ph -> ps )
4	3	spi 1469	. . . 4 |- ( ph -> ps )
5	4	adantr 270	. . 3 |- ( ( ph /\ ch ) -> ps )
6	2, 5	jca 300	. 2 |- ( ( ph /\ ch ) -> ( ch /\ ps ) )
7	1, 6	eximii 1533	1 |- E. x ( ch /\ ps )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1282   E.wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  ferison  2053