Theorem barbari 2138

Description: "Barbari", one of the syllogisms of Aristotelian logic. All ph is ps, all ch is ph, and some ch exist, therefore some ch is ps. (In Aristotelian notation, AAI-1: MaP and SaM therefore SiP.) For example, given "All men are mortal", "All Greeks are men", and "Greeks exist", therefore "Some Greeks are mortal". Note the existence hypothesis (to prove the "some" in the conclusion). Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 30-Aug-2016.)

Hypotheses

Ref	Expression
barbari.maj	|- A. x ( ph -> ps )
barbari.min	|- A. x ( ch -> ph )
barbari.e	|- E. x ch

Assertion

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

Proof of Theorem barbari

Step	Hyp	Ref	Expression
1		barbari.e	. 2 |- E. x ch
2		barbari.maj	. . . . 5 |- A. x ( ph -> ps )
3		barbari.min	. . . . 5 |- A. x ( ch -> ph )
4	2, 3	barbara 2134	. . . 4 |- A. x ( ch -> ps )
5	4	spi 1546	. . 3 |- ( ch -> ps )
6	5	ancli 323	. 2 |- ( ch -> ( ch /\ ps ) )
7	1, 6	eximii 1612	1 |- E. x ( ch /\ ps )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1361   E.wex 1502

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 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-4 1520  ax-ial 1544

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  celaront  2139