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Theorem eqbrrdiv 4575
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 3876 . . 3 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
5 df-br 3876 . . 3 (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
63, 4, 53bitr3g 221 . 2 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
71, 2, 6eqrelrdv 4573 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1299  wcel 1448  cop 3477   class class class wbr 3875  Rel wrel 4482
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-xp 4483  df-rel 4484
This theorem is referenced by: (None)
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