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Theorem dimatis 2403
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 2386 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 . . . . 5  |-  A. x
( ps  ->  ch )
32spi 1799 . . . 4  |-  ( ps 
->  ch )
43adantl 466 . . 3  |-  ( (
ph  /\  ps )  ->  ch )
5 simpl 457 . . 3  |-  ( (
ph  /\  ps )  ->  ph )
64, 5jca 532 . 2  |-  ( (
ph  /\  ps )  ->  ( ch  /\  ph ) )
71, 6eximii 1627 1  |-  E. x
( ch  /\  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1367   E.wex 1586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-12 1792
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1587
This theorem is referenced by: (None)
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