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Theorem hbral 2604
Description: Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by David Abernethy, 13-Dec-2009.)
Hypotheses
Ref Expression
hbral.1  |-  ( y  e.  A  ->  A. x  y  e.  A )
hbral.2  |-  ( ph  ->  A. x ph )
Assertion
Ref Expression
hbral  |-  ( A. y  e.  A  ph  ->  A. x A. y  e.  A  ph )

Proof of Theorem hbral
StepHypRef Expression
1 df-ral 2561 . 2  |-  ( A. y  e.  A  ph  <->  A. y
( y  e.  A  ->  ph ) )
2 hbral.1 . . . 4  |-  ( y  e.  A  ->  A. x  y  e.  A )
3 hbral.2 . . . 4  |-  ( ph  ->  A. x ph )
42, 3hbim 1737 . . 3  |-  ( ( y  e.  A  ->  ph )  ->  A. x
( y  e.  A  ->  ph ) )
54hbal 1722 . 2  |-  ( A. y ( y  e.  A  ->  ph )  ->  A. x A. y ( y  e.  A  ->  ph ) )
61, 5hbxfrbi 1558 1  |-  ( A. y  e.  A  ph  ->  A. x A. y  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530    e. wcel 1696   A.wral 2556
This theorem is referenced by:  tratrbVD  28953
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-ral 2561
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