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Theorem pm10.541 27665
Description: Theorem *10.541 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
pm10.541  |-  ( A. x ( ph  ->  ( ch  \/  ps )
)  <->  ( ch  \/  A. x ( ph  ->  ps ) ) )
Distinct variable group:    ch, x
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem pm10.541
StepHypRef Expression
1 bi2.04 350 . . . 4  |-  ( (
ph  ->  ( -.  ch  ->  ps ) )  <->  ( -.  ch  ->  ( ph  ->  ps ) ) )
21albii 1556 . . 3  |-  ( A. x ( ph  ->  ( -.  ch  ->  ps ) )  <->  A. x
( -.  ch  ->  (
ph  ->  ps ) ) )
3 19.21v 1843 . . 3  |-  ( A. x ( -.  ch  ->  ( ph  ->  ps ) )  <->  ( -.  ch  ->  A. x ( ph  ->  ps ) ) )
42, 3bitri 240 . 2  |-  ( A. x ( ph  ->  ( -.  ch  ->  ps ) )  <->  ( -.  ch  ->  A. x ( ph  ->  ps ) ) )
5 df-or 359 . . . 4  |-  ( ( ch  \/  ps )  <->  ( -.  ch  ->  ps ) )
65imbi2i 303 . . 3  |-  ( (
ph  ->  ( ch  \/  ps ) )  <->  ( ph  ->  ( -.  ch  ->  ps ) ) )
76albii 1556 . 2  |-  ( A. x ( ph  ->  ( ch  \/  ps )
)  <->  A. x ( ph  ->  ( -.  ch  ->  ps ) ) )
8 df-or 359 . 2  |-  ( ( ch  \/  A. x
( ph  ->  ps )
)  <->  ( -.  ch  ->  A. x ( ph  ->  ps ) ) )
94, 7, 83bitr4i 268 1  |-  ( A. x ( ph  ->  ( ch  \/  ps )
)  <->  ( ch  \/  A. x ( ph  ->  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357   A.wal 1530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nf 1535
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