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Theorem eqbrrdiv 5216
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 4652 . . 3 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
5 df-br 4652 . . 3 (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
63, 4, 53bitr3g 302 . 2 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
71, 2, 6eqrelrdv 5214 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1482  wcel 1989  cop 4181   class class class wbr 4651  Rel wrel 5117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-in 3579  df-ss 3586  df-br 4652  df-opab 4711  df-xp 5118  df-rel 5119
This theorem is referenced by:  funcpropd  16554  fullpropd  16574  fthpropd  16575  dvres  23669  eqfunresadj  31645
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