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Theorem cesaro 2389
Description: "Cesaro", one of the syllogisms of Aristotelian logic. No  ph is  ps, all  ch is  ps, and  ch exist, therefore some  ch is not  ph. (In Aristotelian notation, EAO-2: PeM and SaM therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
cesaro.maj  |-  A. x
( ph  ->  -.  ps )
cesaro.min  |-  A. x
( ch  ->  ps )
cesaro.e  |-  E. x ch
Assertion
Ref Expression
cesaro  |-  E. x
( ch  /\  -.  ph )

Proof of Theorem cesaro
StepHypRef Expression
1 cesaro.e . 2  |-  E. x ch
2 cesaro.maj . . . . 5  |-  A. x
( ph  ->  -.  ps )
32spi 1770 . . . 4  |-  ( ph  ->  -.  ps )
4 cesaro.min . . . . 5  |-  A. x
( ch  ->  ps )
54spi 1770 . . . 4  |-  ( ch 
->  ps )
63, 5nsyl3 114 . . 3  |-  ( ch 
->  -.  ph )
76ancli 536 . 2  |-  ( ch 
->  ( ch  /\  -.  ph ) )
81, 7eximii 1588 1  |-  E. x
( ch  /\  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360   A.wal 1550   E.wex 1551
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552
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