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Theorem ralinexa 2834
Description: A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)
Assertion
Ref Expression
ralinexa  |-  ( A. x  e.  A  ( ph  ->  -.  ps )  <->  -. 
E. x  e.  A  ( ph  /\  ps )
)

Proof of Theorem ralinexa
StepHypRef Expression
1 imnan 420 . . 3  |-  ( (
ph  ->  -.  ps )  <->  -.  ( ph  /\  ps ) )
21ralbii 2813 . 2  |-  ( A. x  e.  A  ( ph  ->  -.  ps )  <->  A. x  e.  A  -.  ( ph  /\  ps )
)
3 ralnex 2828 . 2  |-  ( A. x  e.  A  -.  ( ph  /\  ps )  <->  -. 
E. x  e.  A  ( ph  /\  ps )
)
42, 3bitri 249 1  |-  ( A. x  e.  A  ( ph  ->  -.  ps )  <->  -. 
E. x  e.  A  ( ph  /\  ps )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367   A.wral 2732   E.wrex 2733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1621  df-ral 2737  df-rex 2738
This theorem is referenced by:  kmlem7  8449  kmlem13  8455  lspsncv0  17905  ntreq0  19664  lhop1lem  22499  soseq  29499  ltrnnid  36273
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