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Theorem eqrelriiv 5803
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 5802 . 2 ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵)
51, 2, 4mp2an 692 1 𝐴 = 𝐵
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  wcel 2106  cop 4637  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-opab 5211  df-xp 5695  df-rel 5696
This theorem is referenced by:  eqbrriv  5804  inopab  5842  difopab  5843  difopabOLD  5844  inxp  5845  dfres2  6061  restidsing  6073  cnvopab  6160  cnvopabOLD  6161  cnvdif  6166  difxp  6186  cnvcnvsn  6241  dfco2  6267  coiun  6278  co02  6282  coass  6287  ressn  6307  ovoliunlem1  25551  h2hlm  31009  cnvco1  35739  cnvco2  35740  inxprnres  38274  cnviun  43640  coiun1  43642
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