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Theorem otthg 5144
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 4377 . . 3 𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶
2 df-ot 4377 . . 3 𝐷, 𝐸, 𝐹⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹
31, 2eqeq12i 2813 . 2 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹⟩)
4 opex 5123 . . . . 5 𝐴, 𝐵⟩ ∈ V
5 opthg 5136 . . . . 5 ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶𝑊) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹⟩ ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐸⟩ ∧ 𝐶 = 𝐹)))
64, 5mpan 682 . . . 4 (𝐶𝑊 → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹⟩ ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐸⟩ ∧ 𝐶 = 𝐹)))
7 opthg 5136 . . . . . 6 ((𝐴𝑈𝐵𝑉) → (⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐸⟩ ↔ (𝐴 = 𝐷𝐵 = 𝐸)))
87anbi1d 624 . . . . 5 ((𝐴𝑈𝐵𝑉) → ((⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐸⟩ ∧ 𝐶 = 𝐹) ↔ ((𝐴 = 𝐷𝐵 = 𝐸) ∧ 𝐶 = 𝐹)))
9 df-3an 1110 . . . . 5 ((𝐴 = 𝐷𝐵 = 𝐸𝐶 = 𝐹) ↔ ((𝐴 = 𝐷𝐵 = 𝐸) ∧ 𝐶 = 𝐹))
108, 9syl6bbr 281 . . . 4 ((𝐴𝑈𝐵𝑉) → ((⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐸⟩ ∧ 𝐶 = 𝐹) ↔ (𝐴 = 𝐷𝐵 = 𝐸𝐶 = 𝐹)))
116, 10sylan9bbr 507 . . 3 (((𝐴𝑈𝐵𝑉) ∧ 𝐶𝑊) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹⟩ ↔ (𝐴 = 𝐷𝐵 = 𝐸𝐶 = 𝐹)))
12113impa 1137 . 2 ((𝐴𝑈𝐵𝑉𝐶𝑊) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹⟩ ↔ (𝐴 = 𝐷𝐵 = 𝐸𝐶 = 𝐹)))
133, 12syl5bb 275 1 ((𝐴𝑈𝐵𝑉𝐶𝑊) → (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ (𝐴 = 𝐷𝐵 = 𝐸𝐶 = 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  Vcvv 3385  cop 4374  cotp 4376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-ot 4377
This theorem is referenced by:  otsndisj  5175  otiunsndisj  5176  otiunsndisjX  42134
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