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Theorem eqrel2 37679
Description: Equality of relations. (Contributed by Peter Mazsa, 8-Mar-2019.)
Assertion
Ref Expression
eqrel2 ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑦)))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem eqrel2
StepHypRef Expression
1 ssrel3 5779 . . 3 (Rel 𝐴 → (𝐴𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑦)))
2 ssrel3 5779 . . 3 (Rel 𝐵 → (𝐵𝐴 ↔ ∀𝑥𝑦(𝑥𝐵𝑦𝑥𝐴𝑦)))
31, 2bi2anan9 636 . 2 ((Rel 𝐴 ∧ Rel 𝐵) → ((𝐴𝐵𝐵𝐴) ↔ (∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑦) ∧ ∀𝑥𝑦(𝑥𝐵𝑦𝑥𝐴𝑦))))
4 eqss 3992 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 2albiim 1885 . 2 (∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑦) ↔ (∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑦) ∧ ∀𝑥𝑦(𝑥𝐵𝑦𝑥𝐴𝑦)))
63, 4, 53bitr4g 314 1 ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1531   = wceq 1533  wss 3943   class class class wbr 5141  Rel wrel 5674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-in 3950  df-ss 3960  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676
This theorem is referenced by: (None)
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