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| Description: Ordered triple theorem, closed form. (Contributed by Alexander van der Vekens, 10-Mar-2018.) | 
| Ref | Expression | 
|---|---|
| otthg | ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (〈𝐴, 𝐵, 𝐶〉 = 〈𝐷, 𝐸, 𝐹〉 ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-ot 4635 | . . 3 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
| 2 | df-ot 4635 | . . 3 ⊢ 〈𝐷, 𝐸, 𝐹〉 = 〈〈𝐷, 𝐸〉, 𝐹〉 | |
| 3 | 1, 2 | eqeq12i 2755 | . 2 ⊢ (〈𝐴, 𝐵, 𝐶〉 = 〈𝐷, 𝐸, 𝐹〉 ↔ 〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝐷, 𝐸〉, 𝐹〉) | 
| 4 | opex 5469 | . . . . 5 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
| 5 | opthg 5482 | . . . . 5 ⊢ ((〈𝐴, 𝐵〉 ∈ V ∧ 𝐶 ∈ 𝑊) → (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝐷, 𝐸〉, 𝐹〉 ↔ (〈𝐴, 𝐵〉 = 〈𝐷, 𝐸〉 ∧ 𝐶 = 𝐹))) | |
| 6 | 4, 5 | mpan 690 | . . . 4 ⊢ (𝐶 ∈ 𝑊 → (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝐷, 𝐸〉, 𝐹〉 ↔ (〈𝐴, 𝐵〉 = 〈𝐷, 𝐸〉 ∧ 𝐶 = 𝐹))) | 
| 7 | opthg 5482 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → (〈𝐴, 𝐵〉 = 〈𝐷, 𝐸〉 ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸))) | |
| 8 | 7 | anbi1d 631 | . . . . 5 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → ((〈𝐴, 𝐵〉 = 〈𝐷, 𝐸〉 ∧ 𝐶 = 𝐹) ↔ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐸) ∧ 𝐶 = 𝐹))) | 
| 9 | df-3an 1089 | . . . . 5 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹) ↔ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐸) ∧ 𝐶 = 𝐹)) | |
| 10 | 8, 9 | bitr4di 289 | . . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → ((〈𝐴, 𝐵〉 = 〈𝐷, 𝐸〉 ∧ 𝐶 = 𝐹) ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))) | 
| 11 | 6, 10 | sylan9bbr 510 | . . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝐶 ∈ 𝑊) → (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝐷, 𝐸〉, 𝐹〉 ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))) | 
| 12 | 11 | 3impa 1110 | . 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝐷, 𝐸〉, 𝐹〉 ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))) | 
| 13 | 3, 12 | bitrid 283 | 1 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (〈𝐴, 𝐵, 𝐶〉 = 〈𝐷, 𝐸, 𝐹〉 ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 〈cotp 4634 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-ot 4635 | 
| This theorem is referenced by: otsndisj 5524 otiunsndisj 5525 otiunsndisjX 47291 | 
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