ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  darii Unicode version

Theorem darii 2043
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 . . . 4  |-  A. x
( ph  ->  ps )
32spi 1470 . . 3  |-  ( ph  ->  ps )
43anim2i 334 . 2  |-  ( ( ch  /\  ph )  ->  ( ch  /\  ps ) )
51, 4eximii 1534 1  |-  E. x
( ch  /\  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1283   E.wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  ferio  2044
  Copyright terms: Public domain W3C validator