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Theorem calemos 2061
Description: "Calemos", one of the syllogisms of Aristotelian logic. All  ph is  ps (PaM), no  ps is  ch (MeS), and  ch exist, therefore some  ch is not  ph (SoP). (In Aristotelian notation, AEO-4: PaM and MeS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
calemos.maj  |-  A. x
( ph  ->  ps )
calemos.min  |-  A. x
( ps  ->  -.  ch )
calemos.e  |-  E. x ch
Assertion
Ref Expression
calemos  |-  E. x
( ch  /\  -.  ph )

Proof of Theorem calemos
StepHypRef Expression
1 calemos.e . 2  |-  E. x ch
2 calemos.min . . . . . 6  |-  A. x
( ps  ->  -.  ch )
32spi 1470 . . . . 5  |-  ( ps 
->  -.  ch )
43con2i 590 . . . 4  |-  ( ch 
->  -.  ps )
5 calemos.maj . . . . 5  |-  A. x
( ph  ->  ps )
65spi 1470 . . . 4  |-  ( ph  ->  ps )
74, 6nsyl 591 . . 3  |-  ( ch 
->  -.  ph )
87ancli 316 . 2  |-  ( ch 
->  ( ch  /\  -.  ph ) )
91, 8eximii 1534 1  |-  E. x
( ch  /\  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102   A.wal 1283   E.wex 1422
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 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
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