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Theorem otthg 4914
Description: Ordered triple theorem, closed form. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
Assertion
Ref Expression
otthg ((𝐴𝑈𝐵𝑉𝐶𝑊) → (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ (𝐴 = 𝐷𝐵 = 𝐸𝐶 = 𝐹)))

Proof of Theorem otthg
StepHypRef Expression
1 df-ot 4157 . . 3 𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶
2 df-ot 4157 . . 3 𝐷, 𝐸, 𝐹⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹
31, 2eqeq12i 2635 . 2 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹⟩)
4 opex 4893 . . . . 5 𝐴, 𝐵⟩ ∈ V
5 opthg 4906 . . . . 5 ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶𝑊) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹⟩ ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐸⟩ ∧ 𝐶 = 𝐹)))
64, 5mpan 705 . . . 4 (𝐶𝑊 → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹⟩ ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐸⟩ ∧ 𝐶 = 𝐹)))
7 opthg 4906 . . . . . 6 ((𝐴𝑈𝐵𝑉) → (⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐸⟩ ↔ (𝐴 = 𝐷𝐵 = 𝐸)))
87anbi1d 740 . . . . 5 ((𝐴𝑈𝐵𝑉) → ((⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐸⟩ ∧ 𝐶 = 𝐹) ↔ ((𝐴 = 𝐷𝐵 = 𝐸) ∧ 𝐶 = 𝐹)))
9 df-3an 1038 . . . . 5 ((𝐴 = 𝐷𝐵 = 𝐸𝐶 = 𝐹) ↔ ((𝐴 = 𝐷𝐵 = 𝐸) ∧ 𝐶 = 𝐹))
108, 9syl6bbr 278 . . . 4 ((𝐴𝑈𝐵𝑉) → ((⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐸⟩ ∧ 𝐶 = 𝐹) ↔ (𝐴 = 𝐷𝐵 = 𝐸𝐶 = 𝐹)))
116, 10sylan9bbr 736 . . 3 (((𝐴𝑈𝐵𝑉) ∧ 𝐶𝑊) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹⟩ ↔ (𝐴 = 𝐷𝐵 = 𝐸𝐶 = 𝐹)))
12113impa 1256 . 2 ((𝐴𝑈𝐵𝑉𝐶𝑊) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹⟩ ↔ (𝐴 = 𝐷𝐵 = 𝐸𝐶 = 𝐹)))
133, 12syl5bb 272 1 ((𝐴𝑈𝐵𝑉𝐶𝑊) → (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ (𝐴 = 𝐷𝐵 = 𝐸𝐶 = 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3186  cop 4154  cotp 4156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-ot 4157
This theorem is referenced by:  otsndisj  4939  otiunsndisj  4940  otiunsndisjX  40592
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