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Theorem eqrelriv 5743
Description: Inference from extensionality principle for relations. (Contributed by FL, 15-Oct-2012.)
Hypothesis
Ref Expression
eqrelriv.1 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
Assertion
Ref Expression
eqrelriv ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem eqrelriv
StepHypRef Expression
1 eqrelriv.1 . . 3 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
21gen2 1798 . 2 𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
3 eqrel 5738 . 2 ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
42, 3mpbiri 257 1 ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wcel 2106  cop 4590  Rel wrel 5636
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 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3445  df-in 3915  df-ss 3925  df-opab 5166  df-xp 5637  df-rel 5638
This theorem is referenced by:  eqrelriiv  5744  dfrel2  6139  coi1  6212  cnviin  6236
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