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Theorem ralbiim 2680
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 609 . . 3  |-  ( (
ph 
<->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
21ralbii 2567 . 2  |-  ( A. x  e.  A  ( ph 
<->  ps )  <->  A. x  e.  A  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
3 r19.26 2675 . 2  |-  ( A. x  e.  A  (
( ph  ->  ps )  /\  ( ps  ->  ph )
)  <->  ( A. x  e.  A  ( ph  ->  ps )  /\  A. x  e.  A  ( ps  ->  ph ) ) )
42, 3bitri 240 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 176    /\ wa 358   A.wral 2543
This theorem is referenced by:  eqreu  2957  isclo2  16825  chrelat4i  22953  2ralbiim  27952  hlateq  29588
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-nf 1532  df-ral 2548
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