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Theorem calemos 2261
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 1738 . . . . . 6  |-  ( ps 
->  -.  ch )
43con2i 112 . . . . 5  |-  ( ch 
->  -.  ps )
5 calemos.maj . . . . . 6  |-  A. x
( ph  ->  ps )
65spi 1738 . . . . 5  |-  ( ph  ->  ps )
74, 6nsyl 113 . . . 4  |-  ( ch 
->  -.  ph )
87ancli 534 . . 3  |-  ( ch 
->  ( ch  /\  -.  ph ) )
98eximi 1563 . 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 1527   E.wex 1528
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529
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