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Theorem ralbiim 2986
Description: Split a biconditional and distribute quantifier. Restricted quantifier version of albiim 1704. (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 626 . . 3  |-  ( (
ph 
<->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
21ralbii 2885 . 2  |-  ( A. x  e.  A  ( ph 
<->  ps )  <->  A. x  e.  A  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
3 r19.26 2981 . 2  |-  ( A. x  e.  A  (
( ph  ->  ps )  /\  ( ps  ->  ph )
)  <->  ( A. x  e.  A  ( ph  ->  ps )  /\  A. x  e.  A  ( ps  ->  ph ) ) )
42, 3bitri 249 1  |-  ( A. x  e.  A  ( ph 
<->  ps )  <->  ( A. x  e.  A  ( ph  ->  ps )  /\  A. x  e.  A  ( ps  ->  ph ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wral 2804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636
This theorem depends on definitions:  df-bi 185  df-an 369  df-ral 2809
This theorem is referenced by:  eqreu  3288  isclo2  19756  chrelat4i  27490  2ralbiim  32418  hlateq  35520
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