Theorem eqbrrdv 5806

Description: Deduction from extensionality principle for relations. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
eqbrrdv.1	⊢ (𝜑 → Rel 𝐴)
eqbrrdv.2	⊢ (𝜑 → Rel 𝐵)
eqbrrdv.3	⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))

Assertion

Ref	Expression
eqbrrdv	⊢ (𝜑 → 𝐴 = 𝐵)

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

Proof of Theorem eqbrrdv

Step	Hyp	Ref	Expression
1		eqbrrdv.3	. . . 4 ⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))
2		df-br 5149	. . . 4 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
3		df-br 5149	. . . 4 ⊢ (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
4	1, 2, 3	3bitr3g 313	. . 3 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
5	4	alrimivv 1926	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
6		eqbrrdv.1	. . 3 ⊢ (𝜑 → Rel 𝐴)
7		eqbrrdv.2	. . 3 ⊢ (𝜑 → Rel 𝐵)
8		eqrel 5797	. . 3 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
9	6, 7, 8	syl2anc 584	. 2 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
10	5, 9	mpbird 257	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535   = 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:  eqbrrdva  5883  oppcsect2  17827  qusxpid  33371  erimeq2  38660