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Theorem barbari 2257
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 2253 . . . . 5  |-  A. x
( ch  ->  ps )
54spi 1750 . . . 4  |-  ( ch 
->  ps )
65ancli 534 . . 3  |-  ( ch 
->  ( ch  /\  ps ) )
76eximi 1566 . 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 1530   E.wex 1531
This theorem is referenced by:  celaront  2258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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