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Theorem calemos 2399
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 1769 . . . . 5  |-  ( ps 
->  -.  ch )
43con2i 114 . . . 4  |-  ( ch 
->  -.  ps )
5 calemos.maj . . . . 5  |-  A. x
( ph  ->  ps )
65spi 1769 . . . 4  |-  ( ph  ->  ps )
74, 6nsyl 115 . . 3  |-  ( ch 
->  -.  ph )
87ancli 535 . 2  |-  ( ch 
->  ( ch  /\  -.  ph ) )
91, 8eximii 1587 1  |-  E. x
( ch  /\  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   A.wal 1549   E.wex 1550
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551
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