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Theorem dimatis 2123
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 2106 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 1516 . . . 4  |-  ( ps 
->  ch )
43adantl 275 . . 3  |-  ( (
ph  /\  ps )  ->  ch )
5 simpl 108 . . 3  |-  ( (
ph  /\  ps )  ->  ph )
64, 5jca 304 . 2  |-  ( (
ph  /\  ps )  ->  ( ch  /\  ph ) )
71, 6eximii 1582 1  |-  E. x
( ch  /\  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1333   E.wex 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-ial 1514
This theorem depends on definitions:  df-bi 116
This theorem is referenced by: (None)
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