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Theorem felapton 2256
Description: "Felapton", one of the syllogisms of Aristotelian logic. No  ph is  ps, all  ph is  ch, and some  ph exist, therefore some  ch is not  ps. (In Aristotelian notation, EAO-3: MeP and MaS therefore SoP.) For example, "No flowers are animals" and "All flowers are plants", therefore "Some plants are not animals". (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
felapton.maj  |-  A. x
( ph  ->  -.  ps )
felapton.min  |-  A. x
( ph  ->  ch )
felapton.e  |-  E. x ph
Assertion
Ref Expression
felapton  |-  E. x
( ch  /\  -.  ps )

Proof of Theorem felapton
StepHypRef Expression
1 felapton.e . 2  |-  E. x ph
2 felapton.min . . . . 5  |-  A. x
( ph  ->  ch )
32spi 1738 . . . 4  |-  ( ph  ->  ch )
4 felapton.maj . . . . 5  |-  A. x
( ph  ->  -.  ps )
54spi 1738 . . . 4  |-  ( ph  ->  -.  ps )
63, 5jca 518 . . 3  |-  ( ph  ->  ( ch  /\  -.  ps ) )
76eximi 1563 . 2  |-  ( E. x ph  ->  E. x
( ch  /\  -.  ps ) )
81, 7ax-mp 8 1  |-  E. x
( ch  /\  -.  ps )
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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