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Theorem ralinexa 2405
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 659 . . 3  |-  ( (
ph  ->  -.  ps )  <->  -.  ( ph  /\  ps ) )
21ralbii 2384 . 2  |-  ( A. x  e.  A  ( ph  ->  -.  ps )  <->  A. x  e.  A  -.  ( ph  /\  ps )
)
3 ralnex 2369 . 2  |-  ( A. x  e.  A  -.  ( ph  /\  ps )  <->  -. 
E. x  e.  A  ( ph  /\  ps )
)
42, 3bitri 182 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    /\ wa 102    <-> wb 103   A.wral 2359   E.wrex 2360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-5 1381  ax-gen 1383  ax-ie2 1428  ax-4 1445  ax-17 1464
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-ral 2364  df-rex 2365
This theorem is referenced by: (None)
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