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Theorem camestros 2264
Description: "Camestros", one of the syllogisms of Aristotelian logic. All  ph is  ps, no  ch is  ps, and  ch exist, therefore some  ch is not  ph. (In Aristotelian notation, AEO-2: PaM and SeM therefore SoP.) For example, "All horses have hooves", "No humans have hooves", and humans exist, therefore "Some humans are not horses". (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
camestros.maj  |-  A. x
( ph  ->  ps )
camestros.min  |-  A. x
( ch  ->  -.  ps )
camestros.e  |-  E. x ch
Assertion
Ref Expression
camestros  |-  E. x
( ch  /\  -.  ph )

Proof of Theorem camestros
StepHypRef Expression
1 camestros.e . 2  |-  E. x ch
2 camestros.min . . . . . 6  |-  A. x
( ch  ->  -.  ps )
32spi 1750 . . . . 5  |-  ( ch 
->  -.  ps )
4 camestros.maj . . . . . 6  |-  A. x
( ph  ->  ps )
54spi 1750 . . . . 5  |-  ( ph  ->  ps )
63, 5nsyl 113 . . . 4  |-  ( ch 
->  -.  ph )
76ancli 534 . . 3  |-  ( ch 
->  ( ch  /\  -.  ph ) )
87eximi 1566 . 2  |-  ( E. x ch  ->  E. x
( ch  /\  -.  ph ) )
91, 8ax-mp 8 1  |-  E. x
( ch  /\  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1530   E.wex 1531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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