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Theorem eqbrriv 4494
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 3815 . . 3  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
5 df-br 3815 . . 3  |-  ( x B y  <->  <. x ,  y >.  e.  B
)
63, 4, 53bitr3i 208 . 2  |-  ( <.
x ,  y >.  e.  A  <->  <. x ,  y
>.  e.  B )
71, 2, 6eqrelriiv 4493 1  |-  A  =  B
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1287    e. wcel 1436   <.cop 3428   class class class wbr 3814   Rel wrel 4409
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-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-br 3815  df-opab 3869  df-xp 4410  df-rel 4411
This theorem is referenced by:  resco  4892  tpostpos  5964
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