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Theorem nrexralim 2876
Description: Negation of a complex predicate calculus formula. (Contributed by FL, 31-Jul-2009.)
Assertion
Ref Expression
nrexralim  |-  ( -. 
E. x  e.  A  A. y  e.  B  ( ph  ->  ps )  <->  A. x  e.  A  E. y  e.  B  ( ph  /\  -.  ps )
)

Proof of Theorem nrexralim
StepHypRef Expression
1 rexanali 2875 . . 3  |-  ( E. y  e.  B  (
ph  /\  -.  ps )  <->  -. 
A. y  e.  B  ( ph  ->  ps )
)
21ralbii 2834 . 2  |-  ( A. x  e.  A  E. y  e.  B  ( ph  /\  -.  ps )  <->  A. x  e.  A  -.  A. y  e.  B  (
ph  ->  ps ) )
3 ralnex 2846 . 2  |-  ( A. x  e.  A  -.  A. y  e.  B  (
ph  ->  ps )  <->  -.  E. x  e.  A  A. y  e.  B  ( ph  ->  ps ) )
42, 3bitr2i 250 1  |-  ( -. 
E. x  e.  A  A. y  e.  B  ( ph  ->  ps )  <->  A. x  e.  A  E. y  e.  B  ( ph  /\  -.  ps )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369   A.wral 2795   E.wrex 2796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-ral 2800  df-rex 2801
This theorem is referenced by:  bwthOLD  19139
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