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Theorem albiim 1621
Description: Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.)
Assertion
Ref Expression
albiim  |-  ( A. x ( ph  <->  ps )  <->  ( A. x ( ph  ->  ps )  /\  A. x ( ps  ->  ph ) ) )

Proof of Theorem albiim
StepHypRef Expression
1 dfbi2 610 . . 3  |-  ( (
ph 
<->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
21albii 1575 . 2  |-  ( A. x ( ph  <->  ps )  <->  A. x ( ( ph  ->  ps )  /\  ( ps  ->  ph ) ) )
3 19.26 1603 . 2  |-  ( A. x ( ( ph  ->  ps )  /\  ( ps  ->  ph ) )  <->  ( A. x ( ph  ->  ps )  /\  A. x
( ps  ->  ph )
) )
42, 3bitri 241 1  |-  ( A. x ( ph  <->  ps )  <->  ( A. x ( ph  ->  ps )  /\  A. x ( ps  ->  ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549
This theorem is referenced by:  2albiim  1622  equveliOLD  2086  eu1  2302  eqss  3363  ssext  4418  asymref2  5251  pm14.122a  27599  equveliNEW7  29528
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566
This theorem depends on definitions:  df-bi 178  df-an 361
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