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Theorem dimatis 2415
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 2398 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 1865 . . . 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 1659 1  |-  E. x
( ch  /\  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1393   E.wex 1613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-12 1855
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1614
This theorem is referenced by: (None)
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