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Theorem eqbrrdiv 5800
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 5153 . . 3 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
5 df-br 5153 . . 3 (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
63, 4, 53bitr3g 312 . 2 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
71, 2, 6eqrelrdv 5798 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  cop 4638   class class class wbr 5152  Rel wrel 5687
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 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-in 3956  df-ss 3966  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689
This theorem is referenced by:  eqfunresadj  7374  funcpropd  17896  fullpropd  17916  fthpropd  17917  dvres  25860
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