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Theorem hbxfreq 2189
Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1402 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 2149 . 2  |-  ( y  e.  A  <->  y  e.  B )
3 hbxfr.2 . 2  |-  ( y  e.  B  ->  A. x  y  e.  B )
42, 3hbxfrbi 1402 1  |-  ( y  e.  A  ->  A. x  y  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1283    = wceq 1285    e. wcel 1434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-cleq 2076  df-clel 2079
This theorem is referenced by: (None)
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