MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  albiim Unicode version

Theorem albiim 1598
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 1553 . 2  |-  ( A. x ( ph  <->  ps )  <->  A. x ( ( ph  ->  ps )  /\  ( ps  ->  ph ) ) )
3 19.26 1580 . 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 1527
This theorem is referenced by:  2albiim  1599  equveli  1928  eu1  2164  eqss  3194  ssext  4228  asymref2  5060  isconcl7a  26105  pm14.122a  27622  equvelv  29116  a12study3  29135
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544
This theorem depends on definitions:  df-bi 177  df-an 360
  Copyright terms: Public domain W3C validator