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Theorem eqrel2 34411
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 34410 . . 3 (Rel 𝐴 → (𝐴𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑦)))
2 ssrel3 34410 . . 3 (Rel 𝐵 → (𝐵𝐴 ↔ ∀𝑥𝑦(𝑥𝐵𝑦𝑥𝐴𝑦)))
31, 2bi2anan9 620 . 2 ((Rel 𝐴 ∧ Rel 𝐵) → ((𝐴𝐵𝐵𝐴) ↔ (∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑦) ∧ ∀𝑥𝑦(𝑥𝐵𝑦𝑥𝐴𝑦))))
4 eqss 3767 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 2albiim 1969 . 2 (∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑦) ↔ (∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑦) ∧ ∀𝑥𝑦(𝑥𝐵𝑦𝑥𝐴𝑦)))
63, 4, 53bitr4g 303 1 ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1629   = wceq 1631  wss 3723   class class class wbr 4787  Rel wrel 5255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-in 3730  df-ss 3737  df-br 4788  df-opab 4848  df-xp 5256  df-rel 5257
This theorem is referenced by: (None)
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