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Theorem ralinexa 2750
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 412 . . 3  |-  ( (
ph  ->  -.  ps )  <->  -.  ( ph  /\  ps ) )
21ralbii 2729 . 2  |-  ( A. x  e.  A  ( ph  ->  -.  ps )  <->  A. x  e.  A  -.  ( ph  /\  ps )
)
3 ralnex 2715 . 2  |-  ( A. x  e.  A  -.  ( ph  /\  ps )  <->  -. 
E. x  e.  A  ( ph  /\  ps )
)
42, 3bitri 241 1  |-  ( A. x  e.  A  ( ph  ->  -.  ps )  <->  -. 
E. x  e.  A  ( ph  /\  ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   A.wral 2705   E.wrex 2706
This theorem is referenced by:  kmlem7  8036  kmlem13  8042  lspsncv0  16218  ntreq0  17141  lhop1lem  19897  soseq  25529  ltrnnid  30933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-ral 2710  df-rex 2711
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