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Theorem cesare 2391
Description: "Cesare", one of the syllogisms of Aristotelian logic. No  ph is  ps, and all  ch is  ps, therefore no  ch is  ph. (In Aristotelian notation, EAE-2: PeM and SaM therefore SeP.) Related to celarent 2386. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 13-Nov-2016.)
Hypotheses
Ref Expression
cesare.maj  |-  A. x
( ph  ->  -.  ps )
cesare.min  |-  A. x
( ch  ->  ps )
Assertion
Ref Expression
cesare  |-  A. x
( ch  ->  -.  ph )

Proof of Theorem cesare
StepHypRef Expression
1 cesare.maj . . . 4  |-  A. x
( ph  ->  -.  ps )
21spi 1772 . . 3  |-  ( ph  ->  -.  ps )
3 cesare.min . . . 4  |-  A. x
( ch  ->  ps )
43spi 1772 . . 3  |-  ( ch 
->  ps )
52, 4nsyl3 114 . 2  |-  ( ch 
->  -.  ph )
65ax-gen 1556 1  |-  A. x
( ch  ->  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-11 1764
This theorem depends on definitions:  df-bi 179  df-ex 1552
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