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Theorem festino 2248
Description: "Festino", one of the syllogisms of Aristotelian logic. No  ph is  ps, and some  ch is  ps, therefore some  ch is not  ph. (In Aristotelian notation, EIO-2: PeM and SiM therefore SoP.) (Contributed by David A. Wheeler, 25-Nov-2016.)
Hypotheses
Ref Expression
festino.maj  |-  A. x
( ph  ->  -.  ps )
festino.min  |-  E. x
( ch  /\  ps )
Assertion
Ref Expression
festino  |-  E. x
( ch  /\  -.  ph )

Proof of Theorem festino
StepHypRef Expression
1 festino.min . 2  |-  E. x
( ch  /\  ps )
2 festino.maj . . . . . 6  |-  A. x
( ph  ->  -.  ps )
32spi 1738 . . . . 5  |-  ( ph  ->  -.  ps )
43con2i 112 . . . 4  |-  ( ps 
->  -.  ph )
54anim2i 552 . . 3  |-  ( ( ch  /\  ps )  ->  ( ch  /\  -.  ph ) )
65eximi 1563 . 2  |-  ( E. x ( ch  /\  ps )  ->  E. x
( ch  /\  -.  ph ) )
71, 6ax-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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