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Theorem eqbrrdiv 5252
Description: Deduction from extensionality principle for relations. (Contributed by Rodolfo Medina, 10-Oct-2010.)
Hypotheses
Ref Expression
eqbrrdiv.1 Rel 𝐴
eqbrrdiv.2 Rel 𝐵
eqbrrdiv.3 (𝜑 → (𝑥𝐴𝑦𝑥𝐵𝑦))
Assertion
Ref Expression
eqbrrdiv (𝜑𝐴 = 𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦

Proof of Theorem eqbrrdiv
StepHypRef Expression
1 eqbrrdiv.1 . 2 Rel 𝐴
2 eqbrrdiv.2 . 2 Rel 𝐵
3 eqbrrdiv.3 . . 3 (𝜑 → (𝑥𝐴𝑦𝑥𝐵𝑦))
4 df-br 4686 . . 3 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
5 df-br 4686 . . 3 (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
63, 4, 53bitr3g 302 . 2 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
71, 2, 6eqrelrdv 5250 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  cop 4216   class class class wbr 4685  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-in 3614  df-ss 3621  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150
This theorem is referenced by:  funcpropd  16607  fullpropd  16627  fthpropd  16628  dvres  23720  eqfunresadj  31785
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