Theorem celaront 2157

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 2156	1 |- E. x ( ch /\ -. ps )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   A.wal 1371   E.wex 1515

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-ial 1557

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)