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Theorem calemos 2155
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 1546 . . . . 5  |-  ( ps 
->  -.  ch )
43con2i 628 . . . 4  |-  ( ch 
->  -.  ps )
5 calemos.maj . . . . 5  |-  A. x
( ph  ->  ps )
65spi 1546 . . . 4  |-  ( ph  ->  ps )
74, 6nsyl 629 . . 3  |-  ( ch 
->  -.  ph )
87ancli 323 . 2  |-  ( ch 
->  ( ch  /\  -.  ph ) )
91, 8eximii 1612 1  |-  E. x
( ch  /\  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104   A.wal 1361   E.wex 1502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-4 1520  ax-ial 1544
This theorem depends on definitions:  df-bi 117
This theorem is referenced by: (None)
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