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Theorem dimatis 2397
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 2380 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 1769 . . . 4  |-  ( ps 
->  ch )
43adantl 453 . . 3  |-  ( (
ph  /\  ps )  ->  ch )
5 simpl 444 . . 3  |-  ( (
ph  /\  ps )  ->  ph )
64, 5jca 519 . 2  |-  ( (
ph  /\  ps )  ->  ( ch  /\  ph ) )
71, 6eximii 1587 1  |-  E. x
( ch  /\  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1549   E.wex 1550
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551
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