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Theorem calemes 2398
Description: "Calemes", one of the syllogisms of Aristotelian logic. All  ph is  ps, and no  ps is  ch, therefore no  ch is  ph. (In Aristotelian notation, AEE-4: PaM and MeS therefore SeP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
calemes.maj  |-  A. x
( ph  ->  ps )
calemes.min  |-  A. x
( ps  ->  -.  ch )
Assertion
Ref Expression
calemes  |-  A. x
( ch  ->  -.  ph )

Proof of Theorem calemes
StepHypRef Expression
1 calemes.min . . . . 5  |-  A. x
( ps  ->  -.  ch )
21spi 1799 . . . 4  |-  ( ps 
->  -.  ch )
32con2i 120 . . 3  |-  ( ch 
->  -.  ps )
4 calemes.maj . . . 4  |-  A. x
( ph  ->  ps )
54spi 1799 . . 3  |-  ( ph  ->  ps )
63, 5nsyl 121 . 2  |-  ( ch 
->  -.  ph )
76ax-gen 1591 1  |-  A. x
( ch  ->  -.  ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-12 1792
This theorem depends on definitions:  df-bi 185  df-ex 1587
This theorem is referenced by: (None)
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