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Theorem eqbrriv 5784
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 (𝑥𝐴𝑦𝑥𝐵𝑦)
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 (𝑥𝐴𝑦𝑥𝐵𝑦)
4 df-br 5142 . . 3 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
5 df-br 5142 . . 3 (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
63, 4, 53bitr3i 301 . 2 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
71, 2, 6eqrelriiv 5783 1 𝐴 = 𝐵
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wcel 2098  cop 4629   class class class wbr 5141  Rel wrel 5674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-in 3950  df-ss 3960  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676
This theorem is referenced by:  resco  6242  tpostpos  8229  sbthcl  9094  dfle2  13129  dflt2  13130  xpab  35229  idsset  35395  dfbigcup2  35404  imageval  35435  inxpxrn  37776
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