Theorem eqrelrdv2 5761

Description: A version of eqrelrdv 5758. (Contributed by Rodolfo Medina, 10-Oct-2010.)

Hypothesis

Ref	Expression
eqrelrdv2.1	⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))

Assertion

Ref	Expression
eqrelrdv2	⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)

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

Proof of Theorem eqrelrdv2

Step	Hyp	Ref	Expression
1		eqrelrdv2.1	. . . 4 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
2	1	alrimivv 1928	. . 3 ⊢ (𝜑 → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
3		eqrel 5750	. . 3 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
4	2, 3	imbitrrid 246	. 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝜑 → 𝐴 = 𝐵))
5	4	imp 406	1 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ⟨cop 4598  Rel wrel 5646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-ss 3934  df-opab 5173  df-xp 5647  df-rel 5648

This theorem is referenced by:  xpiindi  5802  fliftcnv  7289  dmtpos  8220  ercnv  8695  fpwwe2lem8  10598  invsym2  17732  eqbrrdv2  38863  dibglbN  41167  diclspsn  41195  dih1  41287  dihglbcpreN  41301  dihmeetlem4preN  41307  rfovcnvf1od  44000