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Theorem calemos 2274
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 . . . . . . 7  |-  A. x
( ps  ->  -.  ch )
32spi 1750 . . . . . 6  |-  ( ps 
->  -.  ch )
43con2i 112 . . . . 5  |-  ( ch 
->  -.  ps )
5 calemos.maj . . . . . 6  |-  A. x
( ph  ->  ps )
65spi 1750 . . . . 5  |-  ( ph  ->  ps )
74, 6nsyl 113 . . . 4  |-  ( ch 
->  -.  ph )
87ancli 534 . . 3  |-  ( ch 
->  ( ch  /\  -.  ph ) )
98eximi 1566 . 2  |-  ( E. x ch  ->  E. x
( ch  /\  -.  ph ) )
101, 9ax-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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