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Theorem hbra1 2757
Description:  x is not free in  A. x  e.  A ph. (Contributed by NM, 18-Oct-1996.)
Assertion
Ref Expression
hbra1  |-  ( A. x  e.  A  ph  ->  A. x A. x  e.  A  ph )

Proof of Theorem hbra1
StepHypRef Expression
1 df-ral 2712 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
2 hba1 1805 . 2  |-  ( A. x ( x  e.  A  ->  ph )  ->  A. x A. x ( x  e.  A  ->  ph ) )
31, 2hbxfrbi 1578 1  |-  ( A. x  e.  A  ph  ->  A. x A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550    e. wcel 1726   A.wral 2707
This theorem is referenced by:  hbra2VD  29034  tratrbVD  29035  ssralv2VD  29040  bnj1095  29214  bnj1309  29453
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-11 1762
This theorem depends on definitions:  df-bi 179  df-ex 1552  df-ral 2712
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