Theorem eqbrrdiv 5751

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 5101	. . 3 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
5		df-br 5101	. . 3 ⊢ (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
6	3, 4, 5	3bitr3g 313	. 2 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
7	1, 2, 6	eqrelrdv 5749	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  ⟨cop 4588   class class class wbr 5100  Rel wrel 5637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-ss 3920  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639

This theorem is referenced by:  eqfunresadj  7316  funcpropd  17838  fullpropd  17858  fthpropd  17859  dvres  25880  xpco2  49216  0funcg  49444  0funcALT  49447  functermc2  49868  lmddu  50026  cmddu  50027