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Theorem opideq 36478
Description: Equality conditions for ordered pairs 𝐴, 𝐴 and 𝐵, 𝐵. (Contributed by Peter Mazsa, 22-Jul-2019.) (Revised by Thierry Arnoux, 16-Feb-2022.)
Assertion
Ref Expression
opideq (𝐴𝑉 → (⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐵⟩ ↔ 𝐴 = 𝐵))

Proof of Theorem opideq
StepHypRef Expression
1 opthg 5392 . . 3 ((𝐴𝑉𝐴𝑉) → (⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐵⟩ ↔ (𝐴 = 𝐵𝐴 = 𝐵)))
21anidms 567 . 2 (𝐴𝑉 → (⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐵⟩ ↔ (𝐴 = 𝐵𝐴 = 𝐵)))
3 anidm 565 . 2 ((𝐴 = 𝐵𝐴 = 𝐵) ↔ 𝐴 = 𝐵)
42, 3bitrdi 287 1 (𝐴𝑉 → (⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐵⟩ ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  cop 4567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568
This theorem is referenced by: (None)
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