ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  hbxfreq Unicode version

Theorem hbxfreq 2222
Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1431 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 2182 . 2  |-  ( y  e.  A  <->  y  e.  B )
3 hbxfr.2 . 2  |-  ( y  e.  B  ->  A. x  y  e.  B )
42, 3hbxfrbi 1431 1  |-  ( y  e.  A  ->  A. x  y  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1312    = wceq 1314    e. wcel 1463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-17 1489  ax-ial 1497  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-cleq 2108  df-clel 2111
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator