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Theorem hbral 2783
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 2753 . 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 2015 . . 3  |-  ( ( y  e.  A  ->  ph )  ->  A. x
( y  e.  A  ->  ph ) )
54hbal 1932 . 2  |-  ( A. y ( y  e.  A  ->  ph )  ->  A. x A. y ( y  e.  A  ->  ph ) )
61, 5hbxfrbi 1704 1  |-  ( A. y  e.  A  ph  ->  A. x A. y  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1452    e. wcel 1897   A.wral 2748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-nf 1678  df-ral 2753
This theorem is referenced by:  tratrbVD  37297
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