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Theorem albiim 1417
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 380 . . 3  |-  ( (
ph 
<->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
21albii 1400 . 2  |-  ( A. x ( ph  <->  ps )  <->  A. x ( ( ph  ->  ps )  /\  ( ps  ->  ph ) ) )
3 19.26 1411 . 2  |-  ( A. x ( ( ph  ->  ps )  /\  ( ps  ->  ph ) )  <->  ( A. x ( ph  ->  ps )  /\  A. x
( ps  ->  ph )
) )
42, 3bitri 182 1  |-  ( A. x ( ph  <->  ps )  <->  ( A. x ( ph  ->  ps )  /\  A. x ( ps  ->  ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  2albiim  1418  hbbid  1508  equveli  1683  spsbbi  1766  eu1  1967  eqss  3015  ssext  3978
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