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Theorem pm11.53 1823
Description: Theorem *11.53 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
pm11.53  |-  ( A. x A. y ( ph  ->  ps )  <->  ( E. x ph  ->  A. y ps ) )
Distinct variable groups:    ph, y    ps, x
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem pm11.53
StepHypRef Expression
1 19.21v 1801 . . 3  |-  ( A. y ( ph  ->  ps )  <->  ( ph  ->  A. y ps ) )
21albii 1404 . 2  |-  ( A. x A. y ( ph  ->  ps )  <->  A. x
( ph  ->  A. y ps ) )
3 ax-17 1464 . . . 4  |-  ( ps 
->  A. x ps )
43hbal 1411 . . 3  |-  ( A. y ps  ->  A. x A. y ps )
5419.23h 1432 . 2  |-  ( A. x ( ph  ->  A. y ps )  <->  ( E. x ph  ->  A. y ps ) )
62, 5bitri 182 1  |-  ( A. x A. y ( ph  ->  ps )  <->  ( E. x ph  ->  A. y ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287   E.wex 1426
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 1381  ax-7 1382  ax-gen 1383  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
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