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Theorem alral 2384
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 5 . . 3  |-  ( ph  ->  ( x  e.  A  ->  ph ) )
21alimi 1360 . 2  |-  ( A. x ph  ->  A. x
( x  e.  A  ->  ph ) )
3 df-ral 2328 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
42, 3sylibr 141 1  |-  ( A. x ph  ->  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1257    e. wcel 1409   A.wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354
This theorem depends on definitions:  df-bi 114  df-ral 2328
This theorem is referenced by:  find  4350  findset  10457
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