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Theorem alinexa 1709
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 424 . . 3  |-  ( (
ph  ->  -.  ps )  <->  -.  ( ph  /\  ps ) )
21albii 1688 . 2  |-  ( A. x ( ph  ->  -. 
ps )  <->  A. x  -.  ( ph  /\  ps ) )
3 alnex 1662 . 2  |-  ( A. x  -.  ( ph  /\  ps )  <->  -.  E. x
( ph  /\  ps )
)
42, 3bitri 253 1  |-  ( A. x ( ph  ->  -. 
ps )  <->  -.  E. x
( ph  /\  ps )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371   A.wal 1436   E.wex 1660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1661
This theorem is referenced by:  equs3  1782  ralnex  2872  r2exlem  2949  zfregs2  8220  ac6n  8917  nnunb  10867  alexsubALTlem3  21056  nmobndseqi  26412  bj-exnalimn  31214  frege124d  36219  zfregs2VD  37104
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