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Theorem pm11.53 2024
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 2011 . . 3  |-  ( A. y ( ph  ->  ps )  <->  ( ph  ->  A. y ps ) )
21albii 1554 . 2  |-  ( A. x A. y ( ph  ->  ps )  <->  A. x
( ph  ->  A. y ps ) )
3 nfv 1629 . . . 4  |-  F/ x ps
43nfal 1732 . . 3  |-  F/ x A. y ps
5419.23 1777 . 2  |-  ( A. x ( ph  ->  A. y ps )  <->  ( E. x ph  ->  A. y ps ) )
62, 5bitri 242 1  |-  ( A. x A. y ( ph  ->  ps )  <->  ( E. x ph  ->  A. y ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   A.wal 1532   E.wex 1537
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-17 1628  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538  df-nf 1540
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