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Theorem dimatis 2272
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 2255 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 1750 . . . . 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 1566 . 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 1530   E.wex 1531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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