Theorem celaront 2044

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 2043	1 |- E. x ( ch /\ -. ps )

Syntax hints:   -. wn 3   -> 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: (None)