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Theorem ralim 2783
Description: Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997.)
Assertion
Ref Expression
ralim  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( A. x  e.  A  ph  ->  A. x  e.  A  ps )
)

Proof of Theorem ralim
StepHypRef Expression
1 df-ral 2716 . . 3  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( x  e.  A  ->  ( ph  ->  ps ) ) )
2 ax-2 7 . . . 4  |-  ( ( x  e.  A  -> 
( ph  ->  ps )
)  ->  ( (
x  e.  A  ->  ph )  ->  ( x  e.  A  ->  ps ) ) )
32al2imi 1571 . . 3  |-  ( A. x ( x  e.  A  ->  ( ph  ->  ps ) )  -> 
( A. x ( x  e.  A  ->  ph )  ->  A. x
( x  e.  A  ->  ps ) ) )
41, 3sylbi 189 . 2  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( A. x ( x  e.  A  ->  ph )  ->  A. x
( x  e.  A  ->  ps ) ) )
5 df-ral 2716 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
6 df-ral 2716 . 2  |-  ( A. x  e.  A  ps  <->  A. x ( x  e.  A  ->  ps )
)
74, 5, 63imtr4g 263 1  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( A. x  e.  A  ph  ->  A. x  e.  A  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550    e. wcel 1727   A.wral 2711
This theorem is referenced by:  ral2imi  2788  r19.30  2859  trint  4342  mpteqb  5848  tfrlem1  6665  tz7.49  6731  abianfp  6745  mptelixpg  7128  resixpfo  7129  bnd  7847  kmlem12  8072  lbzbi  10595  r19.29uz  12185  caubnd  12193  alzdvds  12930  ptclsg  17678  isucn2  18340  dfon2lem8  25448  dford3lem2  27136  usgreghash2spot  28556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567
This theorem depends on definitions:  df-bi 179  df-ral 2716
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