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Theorem pm10.55 26932
Description: Theorem *10.55 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
pm10.55  |-  ( ( E. x ( ph  /\ 
ps )  /\  A. x ( ph  ->  ps ) )  <->  ( E. x ph  /\  A. x
( ph  ->  ps )
) )

Proof of Theorem pm10.55
StepHypRef Expression
1 exsimpl 1591 . . 3  |-  ( E. x ( ph  /\  ps )  ->  E. x ph )
21anim1i 554 . 2  |-  ( ( E. x ( ph  /\ 
ps )  /\  A. x ( ph  ->  ps ) )  ->  ( E. x ph  /\  A. x ( ph  ->  ps ) ) )
3 exintr 1616 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  E. x
( ph  /\  ps )
) )
43imdistanri 675 . 2  |-  ( ( E. x ph  /\  A. x ( ph  ->  ps ) )  ->  ( E. x ( ph  /\  ps )  /\  A. x
( ph  ->  ps )
) )
52, 4impbii 182 1  |-  ( ( E. x ( ph  /\ 
ps )  /\  A. x ( ph  ->  ps ) )  <->  ( E. x ph  /\  A. x
( ph  ->  ps )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   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-gen 1536
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538
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