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Theorem alinexa 1565
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 411 . . 3  |-  ( (
ph  ->  -.  ps )  <->  -.  ( ph  /\  ps ) )
21albii 1553 . 2  |-  ( A. x ( ph  ->  -. 
ps )  <->  A. x  -.  ( ph  /\  ps ) )
3 alnex 1530 . 2  |-  ( A. x  -.  ( ph  /\  ps )  <->  -.  E. x
( ph  /\  ps )
)
42, 3bitri 240 1  |-  ( A. x ( ph  ->  -. 
ps )  <->  -.  E. x
( ph  /\  ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528
This theorem is referenced by:  equs3  1625  ralnex  2553  zfregs2  7415  ac6n  8112  nnunb  9961  alexsubALTlem3  17743  nmobndseqi  21357  zfregs2VD  28617
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529
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