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

Theorem barbari 2405
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 . . . . 5  |-  A. x
( ph  ->  ps )
3 barbari.min . . . . 5  |-  A. x
( ch  ->  ph )
42, 3barbara 2401 . . . 4  |-  A. x
( ch  ->  ps )
54spi 1808 . . 3  |-  ( ch 
->  ps )
65ancli 551 . 2  |-  ( ch 
->  ( ch  /\  ps ) )
71, 6eximii 1632 1  |-  E. x
( ch  /\  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1372   E.wex 1591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-12 1798
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1592
This theorem is referenced by:  celaront  2406
  Copyright terms: Public domain W3C validator