MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fresison Unicode version

Theorem fresison 2355
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 448 . . 3  |-  ( ( ps  /\  ch )  ->  ch )
3 fresison.maj . . . . . 6  |-  A. x
( ph  ->  -.  ps )
43spi 1761 . . . . 5  |-  ( ph  ->  -.  ps )
54con2i 114 . . . 4  |-  ( ps 
->  -.  ph )
65adantr 452 . . 3  |-  ( ( ps  /\  ch )  ->  -.  ph )
72, 6jca 519 . 2  |-  ( ( ps  /\  ch )  ->  ( ch  /\  -.  ph ) )
81, 7eximii 1584 1  |-  E. x
( ch  /\  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   A.wal 1546   E.wex 1547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-11 1753
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548
  Copyright terms: Public domain W3C validator