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Theorem eqbrriv 4683
Description: Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.)
Hypotheses
Ref Expression
eqbrriv.1  |-  Rel  A
eqbrriv.2  |-  Rel  B
eqbrriv.3  |-  ( x A y  <->  x B
y )
Assertion
Ref Expression
eqbrriv  |-  A  =  B
Distinct variable groups:    x, y, A   
x, B, y

Proof of Theorem eqbrriv
StepHypRef Expression
1 eqbrriv.1 . 2  |-  Rel  A
2 eqbrriv.2 . 2  |-  Rel  B
3 eqbrriv.3 . . 3  |-  ( x A y  <->  x B
y )
4 df-br 3968 . . 3  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
5 df-br 3968 . . 3  |-  ( x B y  <->  <. x ,  y >.  e.  B
)
63, 4, 53bitr3i 209 . 2  |-  ( <.
x ,  y >.  e.  A  <->  <. x ,  y
>.  e.  B )
71, 2, 6eqrelriiv 4682 1  |-  A  =  B
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1335    e. wcel 2128   <.cop 3564   class class class wbr 3967   Rel wrel 4593
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-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4028  df-xp 4594  df-rel 4595
This theorem is referenced by:  resco  5092  tpostpos  6213
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