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Theorem eqrelriiv 5790
Description: Inference from extensionality principle for relations. (Contributed by NM, 17-Mar-1995.)
Hypotheses
Ref Expression
eqreliiv.1 Rel 𝐴
eqreliiv.2 Rel 𝐵
eqreliiv.3 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
Assertion
Ref Expression
eqrelriiv 𝐴 = 𝐵
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem eqrelriiv
StepHypRef Expression
1 eqreliiv.1 . 2 Rel 𝐴
2 eqreliiv.2 . 2 Rel 𝐵
3 eqreliiv.3 . . 3 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
43eqrelriv 5789 . 2 ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵)
51, 2, 4mp2an 689 1 𝐴 = 𝐵
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1540  wcel 2105  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 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3955  df-ss 3965  df-opab 5211  df-xp 5682  df-rel 5683
This theorem is referenced by:  eqbrriv  5791  inopab  5829  difopab  5830  difopabOLD  5831  dfres2  6041  restidsing  6052  cnvopab  6138  cnvdif  6143  difxp  6163  cnvcnvsn  6218  dfco2  6244  coiun  6255  co02  6259  coass  6264  ressn  6284  ovoliunlem1  25252  h2hlm  30501  cnvco1  35034  cnvco2  35035  inxprnres  37465  cnviun  42704  coiun1  42706
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