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Theorem alral 2479
Description: Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)
Assertion
Ref Expression
alral  |-  ( A. x ph  ->  A. x  e.  A  ph )

Proof of Theorem alral
StepHypRef Expression
1 ax-1 6 . . 3  |-  ( ph  ->  ( x  e.  A  ->  ph ) )
21alimi 1432 . 2  |-  ( A. x ph  ->  A. x
( x  e.  A  ->  ph ) )
3 df-ral 2422 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
42, 3sylibr 133 1  |-  ( A. x ph  ->  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1330    e. wcel 1481   A.wral 2417
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 1424  ax-gen 1426
This theorem depends on definitions:  df-bi 116  df-ral 2422
This theorem is referenced by:  abnex  4372  find  4517  prodeq2w  11353  findset  13297
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