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Theorem eqrelrdv2 5637
 Description: A version of eqrelrdv 5634. (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
StepHypRef Expression
1 eqrelrdv2.1 . . . 4 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
21alrimivv 1929 . . 3 (𝜑 → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
3 eqrel 5627 . . 3 ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
42, 3syl5ibr 249 . 2 ((Rel 𝐴 ∧ Rel 𝐵) → (𝜑𝐴 = 𝐵))
54imp 410 1 (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ⟨cop 4528  Rel wrel 5529 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-in 3865  df-ss 3875  df-opab 5095  df-xp 5530  df-rel 5531 This theorem is referenced by:  xpiindi  5675  fliftcnv  7058  dmtpos  7914  ercnv  8320  fpwwe2lem8  10098  invsym2  17092  eqbrrdv2  36439  dibglbN  38742  diclspsn  38770  dih1  38862  dihglbcpreN  38876  dihmeetlem4preN  38882  rfovcnvf1od  41078
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