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Theorem celarent 2241
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
StepHypRef Expression
1 celarent.maj . 2  |-  A. x
( ph  ->  -.  ps )
2 celarent.min . 2  |-  A. x
( ch  ->  ph )
31, 2barbara 2240 1  |-  A. x
( ch  ->  -.  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544
This theorem depends on definitions:  df-bi 177  df-an 360
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