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Theorem pm10.56 26966
Description: Theorem *10.56 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
pm10.56  |-  ( ( A. x ( ph  ->  ps )  /\  E. x ( ph  /\  ch ) )  ->  E. x
( ps  /\  ch ) )

Proof of Theorem pm10.56
StepHypRef Expression
1 pm3.45 809 . . . . . 6  |-  ( (
ph  ->  ps )  -> 
( ( ph  /\  ch )  ->  ( ps 
/\  ch ) ) )
21con3d 127 . . . . 5  |-  ( (
ph  ->  ps )  -> 
( -.  ( ps 
/\  ch )  ->  -.  ( ph  /\  ch )
) )
32al2imi 1550 . . . 4  |-  ( A. x ( ph  ->  ps )  ->  ( A. x  -.  ( ps  /\  ch )  ->  A. x  -.  ( ph  /\  ch ) ) )
43con3d 127 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( -.  A. x  -.  ( ph  /\ 
ch )  ->  -.  A. x  -.  ( ps 
/\  ch ) ) )
5 df-ex 1531 . . 3  |-  ( E. x ( ph  /\  ch )  <->  -.  A. x  -.  ( ph  /\  ch ) )
6 df-ex 1531 . . 3  |-  ( E. x ( ps  /\  ch )  <->  -.  A. x  -.  ( ps  /\  ch ) )
74, 5, 63imtr4g 263 . 2  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ( ph  /\  ch )  ->  E. x
( ps  /\  ch ) ) )
87imp 420 1  |-  ( ( A. x ( ph  ->  ps )  /\  E. x ( ph  /\  ch ) )  ->  E. x
( ps  /\  ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360   A.wal 1529   E.wex 1530
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1531
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