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Theorem alinexa 1585
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 412 . . 3  |-  ( (
ph  ->  -.  ps )  <->  -.  ( ph  /\  ps ) )
21albii 1572 . 2  |-  ( A. x ( ph  ->  -. 
ps )  <->  A. x  -.  ( ph  /\  ps ) )
3 alnex 1549 . 2  |-  ( A. x  -.  ( ph  /\  ps )  <->  -.  E. x
( ph  /\  ps )
)
42, 3bitri 241 1  |-  ( A. x ( ph  ->  -. 
ps )  <->  -.  E. x
( ph  /\  ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   E.wex 1547
This theorem is referenced by:  equs3  1651  axi11e  2366  ralnex  2659  zfregs2  7602  ac6n  8298  nnunb  10149  alexsubALTlem3  18001  nmobndseqi  22128  zfregs2VD  28294
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548
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