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Theorem hbxfreq 2540
Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1578 for equivalence version. (Contributed by NM, 21-Aug-2007.)
Hypotheses
Ref Expression
hbxfr.1  |-  A  =  B
hbxfr.2  |-  ( y  e.  B  ->  A. x  y  e.  B )
Assertion
Ref Expression
hbxfreq  |-  ( y  e.  A  ->  A. x  y  e.  A )

Proof of Theorem hbxfreq
StepHypRef Expression
1 hbxfr.1 . . 3  |-  A  =  B
21eleq2i 2501 . 2  |-  ( y  e.  A  <->  y  e.  B )
3 hbxfr.2 . 2  |-  ( y  e.  B  ->  A. x  y  e.  B )
42, 3hbxfrbi 1578 1  |-  ( y  e.  A  ->  A. x  y  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550    = wceq 1653    e. wcel 1726
This theorem is referenced by:  bnj1317  29194  bnj1441  29213  bnj1309  29392
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-11 1762  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-cleq 2430  df-clel 2433
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