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Theorem barbari 2140
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 2136 . . . 4  |-  A. x
( ch  ->  ps )
54spi 1547 . . 3  |-  ( ch 
->  ps )
65ancli 323 . 2  |-  ( ch 
->  ( ch  /\  ps ) )
71, 6eximii 1613 1  |-  E. x
( ch  /\  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   A.wal 1362   E.wex 1503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  celaront  2141
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