Theorem eqbrrdiv 5778

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 5111	. . 3 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
5		df-br 5111	. . 3 ⊢ (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
6	3, 4, 5	3bitr3g 316	. 2 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
7	1, 2, 6	eqrelrdv 5776	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567   ∈ wcel 2149  ⟨cop 4597   class class class wbr 5110  Rel wrel 5664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-ss 3930  df-br 5111  df-opab 5175  df-xp 5665  df-rel 5666

This theorem is referenced by:  eqfunresadj  7356  funcpropd  17955  fullpropd  17975  fthpropd  17976  dvres  26035  xpco2  49515  0funcg  49743  0funcALT  49746  functermc2  50167  lmddu  50325  cmddu  50326