Theorem celarent 2302

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 2301	1 |- A.x(ch -> -. ps)

Syntax hints:  -. wn 3   -> wi 4  A.wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-an 360

This theorem is referenced by: (None)