Theorem opideq 38299

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 5497	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐵⟩ ↔ (𝐴 = 𝐵 ∧ 𝐴 = 𝐵)))
2	1	anidms 566	. 2 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐵⟩ ↔ (𝐴 = 𝐵 ∧ 𝐴 = 𝐵)))
3		anidm 564	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐵) ↔ 𝐴 = 𝐵)
4	2, 3	bitrdi 287	1 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐵⟩ ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ⟨cop 4654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655

This theorem is referenced by: (None)