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Theorem fresison 2273
Description: "Fresison", one of the syllogisms of Aristotelian logic. No  ph is  ps (PeM), and some  ps is  ch (MiS), therefore some  ch is not  ph (SoP). (In Aristotelian notation, EIO-4: PeM and MiS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
fresison.maj  |-  A. x
( ph  ->  -.  ps )
fresison.min  |-  E. x
( ps  /\  ch )
Assertion
Ref Expression
fresison  |-  E. x
( ch  /\  -.  ph )

Proof of Theorem fresison
StepHypRef Expression
1 fresison.min . 2  |-  E. x
( ps  /\  ch )
2 simpr 447 . . . 4  |-  ( ( ps  /\  ch )  ->  ch )
3 fresison.maj . . . . . . 7  |-  A. x
( ph  ->  -.  ps )
43spi 1750 . . . . . 6  |-  ( ph  ->  -.  ps )
54con2i 112 . . . . 5  |-  ( ps 
->  -.  ph )
65adantr 451 . . . 4  |-  ( ( ps  /\  ch )  ->  -.  ph )
72, 6jca 518 . . 3  |-  ( ( ps  /\  ch )  ->  ( ch  /\  -.  ph ) )
87eximi 1566 . 2  |-  ( E. x ( ps  /\  ch )  ->  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 1530   E.wex 1531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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