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Theorem exanaliim 1579
Description: A transformation of quantifiers and logical connectives. In classical logic the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.)
Assertion
Ref Expression
exanaliim  |-  ( E. x ( ph  /\  -.  ps )  ->  -.  A. x ( ph  ->  ps ) )

Proof of Theorem exanaliim
StepHypRef Expression
1 annimim 816 . . 3  |-  ( (
ph  /\  -.  ps )  ->  -.  ( ph  ->  ps ) )
21eximi 1532 . 2  |-  ( E. x ( ph  /\  -.  ps )  ->  E. x  -.  ( ph  ->  ps ) )
3 exnalim 1578 . 2  |-  ( E. x  -.  ( ph  ->  ps )  ->  -.  A. x ( ph  ->  ps ) )
42, 3syl 14 1  |-  ( E. x ( ph  /\  -.  ps )  ->  -.  A. x ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> 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-in1 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391
This theorem is referenced by:  rexnalim  2360  nssr  3058  nssssr  3985  brprcneu  5202
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