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Theorem calemos 2157
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 1547 . . . . 5  |-  ( ps 
->  -.  ch )
43con2i 628 . . . 4  |-  ( ch 
->  -.  ps )
5 calemos.maj . . . . 5  |-  A. x
( ph  ->  ps )
65spi 1547 . . . 4  |-  ( ph  ->  ps )
74, 6nsyl 629 . . 3  |-  ( ch 
->  -.  ph )
87ancli 323 . 2  |-  ( ch 
->  ( ch  /\  -.  ph ) )
91, 8eximii 1613 1  |-  E. x
( ch  /\  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104   A.wal 1362   E.wex 1503
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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545
This theorem depends on definitions:  df-bi 117
This theorem is referenced by: (None)
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