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| Description: Equivalence of existence implied by equality of ordered triples. (Contributed by NM, 26-Apr-2015.) | 
| Ref | Expression | 
|---|---|
| oteqex2 | ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝑅, 𝑆〉, 𝑇〉 → (𝐶 ∈ V ↔ 𝑇 ∈ V)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | opeqex 5502 | . 2 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝑅, 𝑆〉, 𝑇〉 → ((〈𝐴, 𝐵〉 ∈ V ∧ 𝐶 ∈ V) ↔ (〈𝑅, 𝑆〉 ∈ V ∧ 𝑇 ∈ V))) | |
| 2 | opex 5468 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
| 3 | 2 | biantrur 530 | . 2 ⊢ (𝐶 ∈ V ↔ (〈𝐴, 𝐵〉 ∈ V ∧ 𝐶 ∈ V)) | 
| 4 | opex 5468 | . . 3 ⊢ 〈𝑅, 𝑆〉 ∈ V | |
| 5 | 4 | biantrur 530 | . 2 ⊢ (𝑇 ∈ V ↔ (〈𝑅, 𝑆〉 ∈ V ∧ 𝑇 ∈ V)) | 
| 6 | 1, 3, 5 | 3bitr4g 314 | 1 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝑅, 𝑆〉, 𝑇〉 → (𝐶 ∈ V ↔ 𝑇 ∈ V)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ 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-ne 2940 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: oteqex 5504 | 
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