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Theorem datisi 2252
Description: "Datisi", one of the syllogisms of Aristotelian logic. All  ph is  ps, and some  ph is  ch, therefore some  ch is  ps. (In Aristotelian notation, AII-3: MaP and MiS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
Hypotheses
Ref Expression
datisi.maj  |-  A. x
( ph  ->  ps )
datisi.min  |-  E. x
( ph  /\  ch )
Assertion
Ref Expression
datisi  |-  E. x
( ch  /\  ps )

Proof of Theorem datisi
StepHypRef Expression
1 datisi.min . 2  |-  E. x
( ph  /\  ch )
2 simpr 447 . . . 4  |-  ( (
ph  /\  ch )  ->  ch )
3 datisi.maj . . . . . 6  |-  A. x
( ph  ->  ps )
43spi 1738 . . . . 5  |-  ( ph  ->  ps )
54adantr 451 . . . 4  |-  ( (
ph  /\  ch )  ->  ps )
62, 5jca 518 . . 3  |-  ( (
ph  /\  ch )  ->  ( ch  /\  ps ) )
76eximi 1563 . 2  |-  ( E. x ( ph  /\  ch )  ->  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 1527   E.wex 1528
This theorem is referenced by:  ferison  2254
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529
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