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Theorem dimatis 2259
Description: "Dimatis", one of the syllogisms of Aristotelian logic. Some  ph is  ps, and all  ps is  ch, therefore some  ch is  ph. (In Aristotelian notation, IAI-4: PiM and MaS therefore SiP.) For example, "Some pets are rabbits.", "All rabbits have fur", therefore "Some fur bearing animals are pets". Like darii 2242 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.)
Hypotheses
Ref Expression
dimatis.maj  |-  E. x
( ph  /\  ps )
dimatis.min  |-  A. x
( ps  ->  ch )
Assertion
Ref Expression
dimatis  |-  E. x
( ch  /\  ph )

Proof of Theorem dimatis
StepHypRef Expression
1 dimatis.maj . 2  |-  E. x
( ph  /\  ps )
2 dimatis.min . . . . . 6  |-  A. x
( ps  ->  ch )
32spi 1738 . . . . 5  |-  ( ps 
->  ch )
43adantl 452 . . . 4  |-  ( (
ph  /\  ps )  ->  ch )
5 simpl 443 . . . 4  |-  ( (
ph  /\  ps )  ->  ph )
64, 5jca 518 . . 3  |-  ( (
ph  /\  ps )  ->  ( ch  /\  ph ) )
76eximi 1563 . 2  |-  ( E. x ( ph  /\  ps )  ->  E. x
( ch  /\  ph ) )
81, 7ax-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
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