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Theorem alinexa 1535
Description: A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)
Assertion
Ref Expression
alinexa  |-  ( A. x ( ph  ->  -. 
ps )  <->  -.  E. x
( ph  /\  ps )
)

Proof of Theorem alinexa
StepHypRef Expression
1 imnan 657 . . 3  |-  ( (
ph  ->  -.  ps )  <->  -.  ( ph  /\  ps ) )
21albii 1400 . 2  |-  ( A. x ( ph  ->  -. 
ps )  <->  A. x  -.  ( ph  /\  ps ) )
3 alnex 1429 . 2  |-  ( A. x  -.  ( ph  /\  ps )  <->  -.  E. x
( ph  /\  ps )
)
42, 3bitri 182 1  |-  ( A. x ( ph  ->  -. 
ps )  <->  -.  E. x
( ph  /\  ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103   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-ie2 1424
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291
This theorem is referenced by:  sbnv  1810  ralnex  2359
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