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

Theorem barbari 2244
Description: "Barbari", one of the syllogisms of Aristotelian logic. All  ph is  ps, all  ch is  ph, and some  ch exist, therefore some  ch is  ps. (In Aristotelian notation, AAI-1: MaP and SaM therefore SiP.) For example, given "All men are mortal", "All Greeks are men", and "Greeks exist", therefore "Some Greeks are mortal". Note the existence hypothesis (to prove the "some" in the conclusion). Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 30-Aug-2016.)
Hypotheses
Ref Expression
barbari.maj  |-  A. x
( ph  ->  ps )
barbari.min  |-  A. x
( ch  ->  ph )
barbari.e  |-  E. x ch
Assertion
Ref Expression
barbari  |-  E. x
( ch  /\  ps )

Proof of Theorem barbari
StepHypRef Expression
1 barbari.e . 2  |-  E. x ch
2 barbari.maj . . . . . 6  |-  A. x
( ph  ->  ps )
3 barbari.min . . . . . 6  |-  A. x
( ch  ->  ph )
42, 3barbara 2240 . . . . 5  |-  A. x
( ch  ->  ps )
54spi 1738 . . . 4  |-  ( ch 
->  ps )
65ancli 534 . . 3  |-  ( ch 
->  ( ch  /\  ps ) )
76eximi 1563 . 2  |-  ( E. x ch  ->  E. x
( ch  /\  ps ) )
81, 7ax-mp 8 1  |-  E. x
( ch  /\  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527   E.wex 1528
This theorem is referenced by:  celaront  2245
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