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Theorem cesaro 2056
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 1474 . . . 4  |-  ( ph  ->  -.  ps )
4 cesaro.min . . . . 5  |-  A. x
( ch  ->  ps )
54spi 1474 . . . 4  |-  ( ch 
->  ps )
63, 5nsyl3 591 . . 3  |-  ( ch 
->  -.  ph )
76ancli 316 . 2  |-  ( ch 
->  ( ch  /\  -.  ph ) )
81, 7eximii 1538 1  |-  E. x
( ch  /\  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102   A.wal 1287   E.wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
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