Theorem eqbrrdv2 36804

Description: Other version of eqbrrdiv 5693. (Contributed by Rodolfo Medina, 30-Sep-2010.)

Hypothesis

Ref	Expression
eqbrrdv2.1	⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))

Assertion

Ref	Expression
eqbrrdv2	⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)

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

Proof of Theorem eqbrrdv2

Step	Hyp	Ref	Expression
1		eqbrrdv2.1	. . . 4 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))
2		df-br 5071	. . . 4 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
3		df-br 5071	. . . 4 ⊢ (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
4	1, 2, 3	3bitr3g 312	. . 3 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
5	4	eqrelrdv2 5694	. 2 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ ((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑)) → 𝐴 = 𝐵)
6	5	anabss5 664	1 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ⟨cop 4564   class class class wbr 5070  Rel wrel 5585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-ss 3900  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587

This theorem is referenced by:  prter3  36823