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Theorem eqbrriv 5086
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 4440 . . 3  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
5 df-br 4440 . . 3  |-  ( x B y  <->  <. x ,  y >.  e.  B
)
63, 4, 53bitr3i 275 . 2  |-  ( <.
x ,  y >.  e.  A  <->  <. x ,  y
>.  e.  B )
71, 2, 6eqrelriiv 5085 1  |-  A  =  B
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1398    e. wcel 1823   <.cop 4022   class class class wbr 4439   Rel wrel 4993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995
This theorem is referenced by:  resco  5494  tpostpos  6967  sbthcl  7632  dfle2  11356  dflt2  11357  idsset  29771  dfbigcup2  29780  imageval  29811
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