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Theorem cesaro 2331
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 1872 . . . 4  |-  ( ph  ->  -.  ps )
4 cesaro.min . . . . 5  |-  A. x
( ch  ->  ps )
54spi 1872 . . . 4  |-  ( ch 
->  ps )
63, 5nsyl3 119 . . 3  |-  ( ch 
->  -.  ph )
76ancli 549 . 2  |-  ( ch 
->  ( ch  /\  -.  ph ) )
81, 7eximii 1666 1  |-  E. x
( ch  /\  -.  ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367   A.wal 1397   E.wex 1620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-12 1862
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1621
This theorem is referenced by: (None)
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