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Theorem pm10.57 36633
Description: Theorem *10.57 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
pm10.57  |-  ( A. x ( ph  ->  ( ps  \/  ch )
)  ->  ( A. x ( ph  ->  ps )  \/  E. x
( ph  /\  ch )
) )

Proof of Theorem pm10.57
StepHypRef Expression
1 alnex 1659 . . . 4  |-  ( A. x  -.  ( ph  /\  ch )  <->  -.  E. x
( ph  /\  ch )
)
2 imnan 423 . . . . . 6  |-  ( (
ph  ->  -.  ch )  <->  -.  ( ph  /\  ch ) )
3 pm2.53 374 . . . . . . . 8  |-  ( ( ps  \/  ch )  ->  ( -.  ps  ->  ch ) )
43con1d 127 . . . . . . 7  |-  ( ( ps  \/  ch )  ->  ( -.  ch  ->  ps ) )
54imim3i 61 . . . . . 6  |-  ( (
ph  ->  ( ps  \/  ch ) )  ->  (
( ph  ->  -.  ch )  ->  ( ph  ->  ps ) ) )
62, 5syl5bir 221 . . . . 5  |-  ( (
ph  ->  ( ps  \/  ch ) )  ->  ( -.  ( ph  /\  ch )  ->  ( ph  ->  ps ) ) )
76al2imi 1681 . . . 4  |-  ( A. x ( ph  ->  ( ps  \/  ch )
)  ->  ( A. x  -.  ( ph  /\  ch )  ->  A. x
( ph  ->  ps )
) )
81, 7syl5bir 221 . . 3  |-  ( A. x ( ph  ->  ( ps  \/  ch )
)  ->  ( -.  E. x ( ph  /\  ch )  ->  A. x
( ph  ->  ps )
) )
98con1d 127 . 2  |-  ( A. x ( ph  ->  ( ps  \/  ch )
)  ->  ( -.  A. x ( ph  ->  ps )  ->  E. x
( ph  /\  ch )
) )
109orrd 379 1  |-  ( A. x ( ph  ->  ( ps  \/  ch )
)  ->  ( A. x ( ph  ->  ps )  \/  E. x
( ph  /\  ch )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 369    /\ wa 370   A.wal 1435   E.wex 1657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-ex 1658
This theorem is referenced by: (None)
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