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Theorem darapti 2270
Description: "Darapti", one of the syllogisms of Aristotelian logic. All  ph is  ps, all  ph is  ch, and some  ph exist, therefore some  ch is  ps. (In Aristotelian notation, AAI-3: MaP and MaS therefore SiP.) For example, "All squares are rectangles" and "All squares are rhombuses", therefore "Some rhombuses are rectangles". (Contributed by David A. Wheeler, 28-Aug-2016.)
Hypotheses
Ref Expression
darapti.maj  |-  A. x
( ph  ->  ps )
darapti.min  |-  A. x
( ph  ->  ch )
darapti.e  |-  E. x ph
Assertion
Ref Expression
darapti  |-  E. x
( ch  /\  ps )

Proof of Theorem darapti
StepHypRef Expression
1 darapti.e . 2  |-  E. x ph
2 darapti.min . . . . 5  |-  A. x
( ph  ->  ch )
32spi 1750 . . . 4  |-  ( ph  ->  ch )
4 darapti.maj . . . . 5  |-  A. x
( ph  ->  ps )
54spi 1750 . . . 4  |-  ( ph  ->  ps )
63, 5jca 518 . . 3  |-  ( ph  ->  ( ch  /\  ps ) )
76eximi 1566 . 2  |-  ( E. x ph  ->  E. x
( ch  /\  ps ) )
81, 7ax-mp 8 1  |-  E. x
( ch  /\  ps )
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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