Users' Mathboxes Mathbox for Andrew Salmon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pm10.55 Unicode version

Theorem pm10.55 27667
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 1582 . . 3  |-  ( E. x ( ph  /\  ps )  ->  E. x ph )
21anim1i 551 . 2  |-  ( ( E. x ( ph  /\ 
ps )  /\  A. x ( ph  ->  ps ) )  ->  ( E. x ph  /\  A. x ( ph  ->  ps ) ) )
3 exintr 1604 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  E. x
( ph  /\  ps )
) )
43imdistanri 672 . 2  |-  ( ( E. x ph  /\  A. x ( ph  ->  ps ) )  ->  ( E. x ( ph  /\  ps )  /\  A. x
( ph  ->  ps )
) )
52, 4impbii 180 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 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
  Copyright terms: Public domain W3C validator