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Theorem eqrelrdv2 5795
Description: A version of eqrelrdv 5792. (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 1931 . . 3 (𝜑 → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
3 eqrel 5784 . . 3 ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
42, 3imbitrrid 245 . 2 ((Rel 𝐴 ∧ Rel 𝐵) → (𝜑𝐴 = 𝐵))
54imp 407 1 (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wcel 2106  cop 4634  Rel wrel 5681
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3955  df-ss 3965  df-opab 5211  df-xp 5682  df-rel 5683
This theorem is referenced by:  xpiindi  5835  fliftcnv  7310  dmtpos  8225  ercnv  8726  fpwwe2lem8  10635  invsym2  17712  eqbrrdv2  37819  dibglbN  40123  diclspsn  40151  dih1  40243  dihglbcpreN  40257  dihmeetlem4preN  40263  rfovcnvf1od  42837
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