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Theorem alinexa 1614
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 691 . . 3  |-  ( (
ph  ->  -.  ps )  <->  -.  ( ph  /\  ps ) )
21albii 1481 . 2  |-  ( A. x ( ph  ->  -. 
ps )  <->  A. x  -.  ( ph  /\  ps ) )
3 alnex 1510 . 2  |-  ( A. x  -.  ( ph  /\  ps )  <->  -.  E. x
( ph  /\  ps )
)
42, 3bitri 184 1  |-  ( A. x ( ph  ->  -. 
ps )  <->  -.  E. x
( ph  /\  ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1362   E.wex 1503
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 615  ax-in2 616  ax-5 1458  ax-gen 1460  ax-ie2 1505
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370
This theorem is referenced by:  sbnv  1900  ralnex  2478
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