Theorem opideq 39415

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 5449	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐵⟩ ↔ (𝐴 = 𝐵 ∧ 𝐴 = 𝐵)))
2	1	anidms 577	. 2 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐵⟩ ↔ (𝐴 = 𝐵 ∧ 𝐴 = 𝐵)))
3		anidm 575	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐵) ↔ 𝐴 = 𝐵)
4	2, 3	bitrdi 290	1 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐵⟩ ↔ 𝐴 = 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2146  ⟨cop 4837

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838

This theorem is used by: (None)