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

Theorem bamalip 2263
Description: "Bamalip", one of the syllogisms of Aristotelian logic. All  ph is  ps, all  ps is  ch, and  ph exist, therefore some  ch is  ph. (In Aristotelian notation, AAI-4: PaM and MaS therefore SiP.) Like barbari 2244. (Contributed by David A. Wheeler, 28-Aug-2016.)
Hypotheses
Ref Expression
bamalip.maj  |-  A. x
( ph  ->  ps )
bamalip.min  |-  A. x
( ps  ->  ch )
bamalip.e  |-  E. x ph
Assertion
Ref Expression
bamalip  |-  E. x
( ch  /\  ph )

Proof of Theorem bamalip
StepHypRef Expression
1 bamalip.e . 2  |-  E. x ph
2 bamalip.maj . . . . . 6  |-  A. x
( ph  ->  ps )
32spi 1738 . . . . 5  |-  ( ph  ->  ps )
4 bamalip.min . . . . . 6  |-  A. x
( ps  ->  ch )
54spi 1738 . . . . 5  |-  ( ps 
->  ch )
63, 5syl 15 . . . 4  |-  ( ph  ->  ch )
76ancri 535 . . 3  |-  ( ph  ->  ( ch  /\  ph ) )
87eximi 1563 . 2  |-  ( E. x ph  ->  E. x
( ch  /\  ph ) )
91, 8ax-mp 8 1  |-  E. x
( ch  /\  ph )
Colors of variables: wff set class
Syntax hints:    -> 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
  Copyright terms: Public domain W3C validator