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

Theorem pm10.55 36688
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 1723 . . 3  |-  ( E. x ( ph  /\  ps )  ->  E. x ph )
21anim1i 570 . 2  |-  ( ( E. x ( ph  /\ 
ps )  /\  A. x ( ph  ->  ps ) )  ->  ( E. x ph  /\  A. x ( ph  ->  ps ) ) )
3 exintr 1750 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  E. x
( ph  /\  ps )
) )
43imdistanri 695 . 2  |-  ( ( E. x ph  /\  A. x ( ph  ->  ps ) )  ->  ( E. x ( ph  /\  ps )  /\  A. x
( ph  ->  ps )
) )
52, 4impbii 190 1  |-  ( ( E. x ( ph  /\ 
ps )  /\  A. x ( ph  ->  ps ) )  <->  ( E. x ph  /\  A. x
( ph  ->  ps )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ 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-an 372  df-ex 1658
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator