Theorem eqbrrdiv 5807

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 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))

Assertion

Ref	Expression
eqbrrdiv	⊢ (𝜑 → 𝐴 = 𝐵)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦

Proof of Theorem eqbrrdiv

Step	Hyp	Ref	Expression
1		eqbrrdiv.1	. 2 ⊢ Rel 𝐴
2		eqbrrdiv.2	. 2 ⊢ Rel 𝐵
3		eqbrrdiv.3	. . 3 ⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))
4		df-br 5149	. . 3 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
5		df-br 5149	. . 3 ⊢ (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
6	3, 4, 5	3bitr3g 313	. 2 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
7	1, 2, 6	eqrelrdv 5805	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2106  ⟨cop 4637   class class class wbr 5148  Rel wrel 5694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696

This theorem is referenced by:  eqfunresadj  7380  funcpropd  17954  fullpropd  17974  fthpropd  17975  dvres  25961