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| Description: Ordered triple theorem, with triple expressed with ordered pairs. (Contributed by NM, 1-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.) | 
| Ref | Expression | 
|---|---|
| otth.1 | ⊢ 𝐴 ∈ V | 
| otth.2 | ⊢ 𝐵 ∈ V | 
| otth.3 | ⊢ 𝑅 ∈ V | 
| Ref | Expression | 
|---|---|
| otth2 | ⊢ (〈〈𝐴, 𝐵〉, 𝑅〉 = 〈〈𝐶, 𝐷〉, 𝑆〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | otth.1 | . . . 4 ⊢ 𝐴 ∈ V | |
| 2 | otth.2 | . . . 4 ⊢ 𝐵 ∈ V | |
| 3 | 1, 2 | opth 5481 | . . 3 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | 
| 4 | 3 | anbi1i 624 | . 2 ⊢ ((〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ∧ 𝑅 = 𝑆) ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∧ 𝑅 = 𝑆)) | 
| 5 | opex 5469 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
| 6 | otth.3 | . . 3 ⊢ 𝑅 ∈ V | |
| 7 | 5, 6 | opth 5481 | . 2 ⊢ (〈〈𝐴, 𝐵〉, 𝑅〉 = 〈〈𝐶, 𝐷〉, 𝑆〉 ↔ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ∧ 𝑅 = 𝑆)) | 
| 8 | df-3an 1089 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆) ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∧ 𝑅 = 𝑆)) | |
| 9 | 4, 7, 8 | 3bitr4i 303 | 1 ⊢ (〈〈𝐴, 𝐵〉, 𝑅〉 = 〈〈𝐶, 𝐷〉, 𝑆〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 | 
| 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 | 
| This theorem is referenced by: otth 5489 oprabidw 7462 oprabid 7463 eloprabga 7542 | 
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