Theorem celarent 2113

Description: "Celarent", one of the syllogisms of Aristotelian logic. No ph is ps, and all ch is ph, therefore no ch is ps. (In Aristotelian notation, EAE-1: MeP and SaM therefore SeP.) For example, given the "No reptiles have fur" and "All snakes are reptiles", therefore "No snakes have fur". Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)

Hypotheses

Ref	Expression
celarent.maj	|- A. x ( ph -> -. ps )
celarent.min	|- A. x ( ch -> ph )

Assertion

Ref	Expression
celarent	|- A. x ( ch -> -. ps )

Proof of Theorem celarent

Step	Hyp	Ref	Expression
1		celarent.maj	. 2 |- A. x ( ph -> -. ps )
2		celarent.min	. 2 |- A. x ( ch -> ph )
3	1, 2	barbara 2112	1 |- A. x ( ch -> -. ps )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1341

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 1435  ax-gen 1437

This theorem is referenced by: (None)