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Theorem eqbrriv 5355
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 4787 . . 3 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
5 df-br 4787 . . 3 (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
63, 4, 53bitr3i 290 . 2 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
71, 2, 6eqrelriiv 5354 1 𝐴 = 𝐵
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1631  wcel 2145  cop 4322   class class class wbr 4786  Rel wrel 5254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-in 3730  df-ss 3737  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256
This theorem is referenced by:  resco  5783  tpostpos  7524  sbthcl  8238  dfle2  12185  dflt2  12186  idsset  32334  dfbigcup2  32343  imageval  32374  inxpxrn  34495
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