Theorem eqbrrdv 4852

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 4115	. . . 4 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
3		df-br 4115	. . . 4 ⊢ (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
4	1, 2, 3	3bitr3g 222	. . 3 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
5	4	alrimivv 1924	. 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 4844	. . 3 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
9	6, 7, 8	syl2anc 411	. 2 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
10	5, 9	mpbird 167	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1396   = wceq 1398   ∈ wcel 2205  ⟨cop 3697   class class class wbr 4114  Rel wrel 4759

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-xp 4760  df-rel 4761

This theorem is referenced by:  eqbrrdva  4930