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Theorem eqrelriv 5358
 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 1860 . 2 𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
3 eqrel 5354 . 2 ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
42, 3mpbiri 248 1 ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1618   = wceq 1620   ∈ wcel 2127  ⟨cop 4315  Rel wrel 5259 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-in 3710  df-ss 3717  df-opab 4853  df-xp 5260  df-rel 5261 This theorem is referenced by:  eqrelriiv  5359  dfrel2  5729  coi1  5800  cnviin  5821
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