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Theorem negcmpprcal1 25048
Description: Negation of a complex predicate calculus formula. (Contributed by FL, 31-Jul-2009.)
Assertion
Ref Expression
negcmpprcal1  |-  ( -. 
E. x  e.  A  A. y  e.  B  ( ph  ->  ps )  <->  A. x  e.  A  E. y  e.  B  ( ph  /\  -.  ps )
)

Proof of Theorem negcmpprcal1
StepHypRef Expression
1 rexanali 2602 . . 3  |-  ( E. y  e.  B  (
ph  /\  -.  ps )  <->  -. 
A. y  e.  B  ( ph  ->  ps )
)
21ralbii 2580 . 2  |-  ( A. x  e.  A  E. y  e.  B  ( ph  /\  -.  ps )  <->  A. x  e.  A  -.  A. y  e.  B  (
ph  ->  ps ) )
3 ralnex 2566 . 2  |-  ( A. x  e.  A  -.  A. y  e.  B  (
ph  ->  ps )  <->  -.  E. x  e.  A  A. y  e.  B  ( ph  ->  ps ) )
42, 3bitr2i 241 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wral 2556   E.wrex 2557
This theorem is referenced by:  bwt2  25695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-ral 2561  df-rex 2562
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