Theorem eqrelrdv2 5786

Description: A version of eqrelrdv 5783. (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 1961	. . 3 ⊢ (𝜑 → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
3		eqrel 5775	. . 3 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
4	2, 3	imbitrrid 249	. 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝜑 → 𝐴 = 𝐵))
5	4	imp 412	1 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2146  ⟨cop 4600  Rel wrel 5671

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-ss 3925  df-opab 5179  df-xp 5672  df-rel 5673

This theorem is used by:  xpiindi  5826  fliftcnv  7320  dmtpos  8243  ercnv  8725  fpwwe2lem8  10641  invsym2  17845  ecxrn2  39098  eqbrrdv2  39678  dibglbN  41981  diclspsn  42009  dih1  42101  dihglbcpreN  42115  dihmeetlem4preN  42121  rfovcnvf1od  44771