Theorem datisi 2155

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 110	. . 3 |- ( ( ph /\ ch ) -> ch )
3		datisi.maj	. . . . 5 |- A. x ( ph -> ps )
4	3	spi 1550	. . . 4 |- ( ph -> ps )
5	4	adantr 276	. . 3 |- ( ( ph /\ ch ) -> ps )
6	2, 5	jca 306	. 2 |- ( ( ph /\ ch ) -> ( ch /\ ps ) )
7	1, 6	eximii 1616	1 |- E. x ( ch /\ ps )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1362   E.wex 1506

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  ferison  2157