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Theorem eqrelrdv 5630
 Description: Deduce equality of relations from equivalence of membership. (Contributed by Rodolfo Medina, 10-Oct-2010.)
Hypotheses
Ref Expression
eqrelrdv.1 Rel 𝐴
eqrelrdv.2 Rel 𝐵
eqrelrdv.3 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
Assertion
Ref Expression
eqrelrdv (𝜑𝐴 = 𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦

Proof of Theorem eqrelrdv
StepHypRef Expression
1 eqrelrdv.3 . . 3 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
21alrimivv 1929 . 2 (𝜑 → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
3 eqrelrdv.1 . . 3 Rel 𝐴
4 eqrelrdv.2 . . 3 Rel 𝐵
5 eqrel 5623 . . 3 ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
63, 4, 5mp2an 691 . 2 (𝐴 = 𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
72, 6sylibr 237 1 (𝜑𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ⟨cop 4531  Rel wrel 5525 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 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-ss 3898  df-opab 5094  df-xp 5526  df-rel 5527 This theorem is referenced by:  eqbrrdiv  5632  fcnvres  6531  fmptco  6869  fpwwe2lem7  10051  fpwwe2lem11  10055  fsumcom2  15124  fprodcom2  15333  gsumcom2  19092  lgsquadlem1  25974  lgsquadlem2  25975  fmptcof2  30430  dfcnv2  30449  dih1dimatlem  38644
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