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| Description: An ordered pair commutes iff its members are equal. (Contributed by NM, 28-May-2009.) | 
| Ref | Expression | 
|---|---|
| opcom.1 | ⊢ 𝐴 ∈ V | 
| opcom.2 | ⊢ 𝐵 ∈ V | 
| Ref | Expression | 
|---|---|
| opcom | ⊢ (〈𝐴, 𝐵〉 = 〈𝐵, 𝐴〉 ↔ 𝐴 = 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | opcom.1 | . . 3 ⊢ 𝐴 ∈ V | |
| 2 | opcom.2 | . . 3 ⊢ 𝐵 ∈ V | |
| 3 | 1, 2 | opth 5480 | . 2 ⊢ (〈𝐴, 𝐵〉 = 〈𝐵, 𝐴〉 ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐴)) | 
| 4 | eqcom 2743 | . . 3 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | |
| 5 | 4 | anbi2i 623 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐴) ↔ (𝐴 = 𝐵 ∧ 𝐴 = 𝐵)) | 
| 6 | anidm 564 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐵) ↔ 𝐴 = 𝐵) | |
| 7 | 3, 5, 6 | 3bitri 297 | 1 ⊢ (〈𝐴, 𝐵〉 = 〈𝐵, 𝐴〉 ↔ 𝐴 = 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 〈cop 4631 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 | 
| This theorem is referenced by: (None) | 
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