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Theorem albiim 1601
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 609 . . 3  |-  ( (
ph 
<->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
21albii 1556 . 2  |-  ( A. x ( ph  <->  ps )  <->  A. x ( ( ph  ->  ps )  /\  ( ps  ->  ph ) ) )
3 19.26 1583 . 2  |-  ( A. x ( ( ph  ->  ps )  /\  ( ps  ->  ph ) )  <->  ( A. x ( ph  ->  ps )  /\  A. x
( ps  ->  ph )
) )
42, 3bitri 240 1  |-  ( A. x ( ph  <->  ps )  <->  ( A. x ( ph  ->  ps )  /\  A. x ( ps  ->  ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530
This theorem is referenced by:  2albiim  1602  equveli  1941  eu1  2177  eqss  3207  ssext  4244  asymref2  5076  isconcl7a  26208  pm14.122a  27725  equveliNEW7  29503  equvelv  29738  a12study3  29757
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547
This theorem depends on definitions:  df-bi 177  df-an 360
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