Theorem opideq 36101

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	⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐵⟩ ↔ 𝐴 = 𝐵))

Proof of Theorem opideq

Step	Hyp	Ref	Expression
1		opthg 5335	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐵⟩ ↔ (𝐴 = 𝐵 ∧ 𝐴 = 𝐵)))
2	1	anidms 570	. 2 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐵⟩ ↔ (𝐴 = 𝐵 ∧ 𝐴 = 𝐵)))
3		anidm 568	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐵) ↔ 𝐴 = 𝐵)
4	2, 3	bitrdi 290	1 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐵⟩ ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1542   ∈ wcel 2114  ⟨cop 4522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3400  df-dif 3846  df-un 3848  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523

This theorem is referenced by: (None)