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Theorem fesapo 2262
Description: "Fesapo", one of the syllogisms of Aristotelian logic. No  ph is  ps, all  ps is  ch, and  ps exist, therefore some  ch is not  ph. (In Aristotelian notation, EAO-4: PeM and MaS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
fesapo.maj  |-  A. x
( ph  ->  -.  ps )
fesapo.min  |-  A. x
( ps  ->  ch )
fesapo.e  |-  E. x ps
Assertion
Ref Expression
fesapo  |-  E. x
( ch  /\  -.  ph )

Proof of Theorem fesapo
StepHypRef Expression
1 fesapo.e . 2  |-  E. x ps
2 fesapo.min . . . . 5  |-  A. x
( ps  ->  ch )
32spi 1738 . . . 4  |-  ( ps 
->  ch )
4 fesapo.maj . . . . . 6  |-  A. x
( ph  ->  -.  ps )
54spi 1738 . . . . 5  |-  ( ph  ->  -.  ps )
65con2i 112 . . . 4  |-  ( ps 
->  -.  ph )
73, 6jca 518 . . 3  |-  ( ps 
->  ( ch  /\  -.  ph ) )
87eximi 1563 . 2  |-  ( E. x ps  ->  E. x
( ch  /\  -.  ph ) )
91, 8ax-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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