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Theorem ralbiim 2492
Description: Split a biconditional and distribute quantifier. (Contributed by NM, 3-Jun-2012.)
Assertion
Ref Expression
ralbiim  |-  ( A. x  e.  A  ( ph 
<->  ps )  <->  ( A. x  e.  A  ( ph  ->  ps )  /\  A. x  e.  A  ( ps  ->  ph ) ) )

Proof of Theorem ralbiim
StepHypRef Expression
1 dfbi2 380 . . 3  |-  ( (
ph 
<->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
21ralbii 2373 . 2  |-  ( A. x  e.  A  ( ph 
<->  ps )  <->  A. x  e.  A  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
3 r19.26 2486 . 2  |-  ( A. x  e.  A  (
( ph  ->  ps )  /\  ( ps  ->  ph )
)  <->  ( A. x  e.  A  ( ph  ->  ps )  /\  A. x  e.  A  ( ps  ->  ph ) ) )
42, 3bitri 182 1  |-  ( A. x  e.  A  ( ph 
<->  ps )  <->  ( A. x  e.  A  ( ph  ->  ps )  /\  A. x  e.  A  ( ps  ->  ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wral 2349
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-5 1377  ax-gen 1379  ax-4 1441  ax-17 1460
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-ral 2354
This theorem is referenced by:  eqreu  2785
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