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Theorem ralinexa 2781
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 422 . . 3  |-  ( (
ph  ->  -.  ps )  <->  -.  ( ph  /\  ps ) )
21ralbii 2760 . 2  |-  ( A. x  e.  A  ( ph  ->  -.  ps )  <->  A. x  e.  A  -.  ( ph  /\  ps )
)
3 ralnex 2746 . 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 369   A.wral 2736   E.wrex 2737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-12 1792
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-ral 2741  df-rex 2742
This theorem is referenced by:  kmlem7  8346  kmlem13  8352  lspsncv0  17249  ntreq0  18703  lhop1lem  21507  soseq  27737  ltrnnid  33876
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