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Theorem ralim 2782
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 2715 . . 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 1606 . . 3  |-  ( A. x ( x  e.  A  ->  ( ph  ->  ps ) )  -> 
( A. x ( x  e.  A  ->  ph )  ->  A. x
( x  e.  A  ->  ps ) ) )
41, 3sylbi 195 . 2  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( A. x ( x  e.  A  ->  ph )  ->  A. x
( x  e.  A  ->  ps ) ) )
5 df-ral 2715 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
6 df-ral 2715 . 2  |-  ( A. x  e.  A  ps  <->  A. x ( x  e.  A  ->  ps )
)
74, 5, 63imtr4g 270 1  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( A. x  e.  A  ph  ->  A. x  e.  A  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1367    e. wcel 1756   A.wral 2710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602
This theorem depends on definitions:  df-bi 185  df-ral 2715
This theorem is referenced by:  ral2imi  2787  r19.30  2860  trint  4395  mpteqb  5783  tz7.49  6892  mptelixpg  7292  resixpfo  7293  bnd  8091  kmlem12  8322  lbzbi  10935  r19.29uz  12830  caubnd  12838  alzdvds  13575  ptclsg  19168  isucn2  19834  dfon2lem8  27572  dford3lem2  29347  usgreghash2spot  30633
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