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Theorem ralim 2529
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 2453 . . 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 1451 . . 3  |-  ( A. x ( x  e.  A  ->  ( ph  ->  ps ) )  -> 
( A. x ( x  e.  A  ->  ph )  ->  A. x
( x  e.  A  ->  ps ) ) )
41, 3sylbi 120 . 2  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( A. x ( x  e.  A  ->  ph )  ->  A. x
( x  e.  A  ->  ps ) ) )
5 df-ral 2453 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
6 df-ral 2453 . 2  |-  ( A. x  e.  A  ps  <->  A. x ( x  e.  A  ->  ps )
)
74, 5, 63imtr4g 204 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 1346    e. wcel 2141   A.wral 2448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442
This theorem depends on definitions:  df-bi 116  df-ral 2453
This theorem is referenced by:  ral2imi  2535  trint  4100  peano2  4577  mpteqb  5584  mptelixpg  6708  lbzbi  9562  r19.29uz  10943  alzdvds  11801
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