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

Theorem fesapo 2406
Description: "Fesapo", one of the syllogisms of Aristotelian logic. No  ph is  ps, all  ps is  ch, and  ps exist, therefore some  ch is not  ph. (In Aristotelian notation, EAO-4: PeM and MaS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
fesapo.maj  |-  A. x
( ph  ->  -.  ps )
fesapo.min  |-  A. x
( ps  ->  ch )
fesapo.e  |-  E. x ps
Assertion
Ref Expression
fesapo  |-  E. x
( ch  /\  -.  ph )

Proof of Theorem fesapo
StepHypRef Expression
1 fesapo.e . 2  |-  E. x ps
2 fesapo.min . . . 4  |-  A. x
( ps  ->  ch )
32spi 1771 . . 3  |-  ( ps 
->  ch )
4 fesapo.maj . . . . 5  |-  A. x
( ph  ->  -.  ps )
54spi 1771 . . . 4  |-  ( ph  ->  -.  ps )
65con2i 115 . . 3  |-  ( ps 
->  -.  ph )
73, 6jca 520 . 2  |-  ( ps 
->  ( ch  /\  -.  ph ) )
81, 7eximii 1588 1  |-  E. x
( ch  /\  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360   A.wal 1550   E.wex 1551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-11 1763
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552
  Copyright terms: Public domain W3C validator