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Theorem eqbrriv 5752
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 5111 . . 3 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
5 df-br 5111 . . 3 (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
63, 4, 53bitr3i 301 . 2 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
71, 2, 6eqrelriiv 5751 1 𝐴 = 𝐵
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  wcel 2107  cop 4597   class class class wbr 5110  Rel wrel 5643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3450  df-in 3922  df-ss 3932  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645
This theorem is referenced by:  resco  6207  tpostpos  8182  sbthcl  9046  dfle2  13073  dflt2  13074  xpab  34337  idsset  34504  dfbigcup2  34513  imageval  34544  inxpxrn  36886
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