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Theorem ralbiim 2845
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 611 . . 3  |-  ( (
ph 
<->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
21ralbii 2731 . 2  |-  ( A. x  e.  A  ( ph 
<->  ps )  <->  A. x  e.  A  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
3 r19.26 2840 . 2  |-  ( A. x  e.  A  (
( ph  ->  ps )  /\  ( ps  ->  ph )
)  <->  ( A. x  e.  A  ( ph  ->  ps )  /\  A. x  e.  A  ( ps  ->  ph ) ) )
42, 3bitri 242 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    <-> wb 178    /\ wa 360   A.wral 2707
This theorem is referenced by:  eqreu  3128  isclo2  17154  chrelat4i  23878  2ralbiim  27930  hlateq  30258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-ral 2712
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