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Theorem darii 2242
Description: "Darii", one of the syllogisms of Aristotelian logic. All  ph is  ps, and some  ch is  ph, therefore some  ch is  ps. (In Aristotelian notation, AII-1: MaP and SiM therefore SiP.) For example, given "All rabbits have fur" and "Some pets are rabbits", therefore "Some pets have fur". Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.)
Hypotheses
Ref Expression
darii.maj  |-  A. x
( ph  ->  ps )
darii.min  |-  E. x
( ch  /\  ph )
Assertion
Ref Expression
darii  |-  E. x
( ch  /\  ps )

Proof of Theorem darii
StepHypRef Expression
1 darii.min . 2  |-  E. x
( ch  /\  ph )
2 darii.maj . . . . 5  |-  A. x
( ph  ->  ps )
32spi 1738 . . . 4  |-  ( ph  ->  ps )
43anim2i 552 . . 3  |-  ( ( ch  /\  ph )  ->  ( ch  /\  ps ) )
54eximi 1563 . 2  |-  ( E. x ( ch  /\  ph )  ->  E. x
( ch  /\  ps ) )
61, 5ax-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:  ferio  2243
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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