Theorem celaront 2102

Description: "Celaront", one of the syllogisms of Aristotelian logic. No ph is ps, all ch is ph, and some ch exist, therefore some ch is not ps. (In Aristotelian notation, EAO-1: MeP and SaM therefore SoP.) For example, given "No reptiles have fur", "All snakes are reptiles.", and "Snakes exist.", prove "Some snakes have no fur". Note the existence hypothesis. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)

Hypotheses

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

Assertion

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

Proof of Theorem celaront

Step	Hyp	Ref	Expression
1		celaront.maj	. 2 |- A. x ( ph -> -. ps )
2		celaront.min	. 2 |- A. x ( ch -> ph )
3		celaront.e	. 2 |- E. x ch
4	1, 2, 3	barbari 2101	1 |- E. x ( ch /\ -. ps )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   A.wal 1329   E.wex 1468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)