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Theorem exanaliim 1647
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 686 . . 3  |-  ( (
ph  /\  -.  ps )  ->  -.  ( ph  ->  ps ) )
21eximi 1600 . 2  |-  ( E. x ( ph  /\  -.  ps )  ->  E. x  -.  ( ph  ->  ps ) )
3 exnalim 1646 . 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 104   A.wal 1351   E.wex 1492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461
This theorem is referenced by:  rexnalim  2466  nssr  3215  nssssr  4222  brprcneu  5508
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