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Theorem hbxfreq 2386
Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1555 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 2347 . 2  |-  ( y  e.  A  <->  y  e.  B )
3 hbxfr.2 . 2  |-  ( y  e.  B  ->  A. x  y  e.  B )
42, 3hbxfrbi 1555 1  |-  ( y  e.  A  ->  A. x  y  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527    = wceq 1623    e. wcel 1684
This theorem is referenced by:  bnj1317  28854  bnj1441  28873  bnj1309  29052
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-cleq 2276  df-clel 2279
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