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Theorem ralim 2614
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 2548 . . 3  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( x  e.  A  ->  ( ph  ->  ps ) ) )
2 ax-2 6 . . . 4  |-  ( ( x  e.  A  -> 
( ph  ->  ps )
)  ->  ( (
x  e.  A  ->  ph )  ->  ( x  e.  A  ->  ps ) ) )
32al2imi 1548 . . 3  |-  ( A. x ( x  e.  A  ->  ( ph  ->  ps ) )  -> 
( A. x ( x  e.  A  ->  ph )  ->  A. x
( x  e.  A  ->  ps ) ) )
41, 3sylbi 187 . 2  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( A. x ( x  e.  A  ->  ph )  ->  A. x
( x  e.  A  ->  ps ) ) )
5 df-ral 2548 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
6 df-ral 2548 . 2  |-  ( A. x  e.  A  ps  <->  A. x ( x  e.  A  ->  ps )
)
74, 5, 63imtr4g 261 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 1527    e. wcel 1684   A.wral 2543
This theorem is referenced by:  ral2imi  2619  r19.30  2685  trint  4128  mpteqb  5614  tfrlem1  6391  tz7.49  6457  abianfp  6471  mptelixpg  6853  resixpfo  6854  bnd  7562  kmlem12  7787  lbzbi  10306  r19.29uz  11834  caubnd  11842  alzdvds  12578  ptclsg  17309  dfon2lem8  24146  dford3lem2  27120  stoweid  27812
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544
This theorem depends on definitions:  df-bi 177  df-ral 2548
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