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Theorem otthg 5058
 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 4294 . . 3 𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶
2 df-ot 4294 . . 3 𝐷, 𝐸, 𝐹⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹
31, 2eqeq12i 2738 . 2 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹⟩)
4 opex 5037 . . . . 5 𝐴, 𝐵⟩ ∈ V
5 opthg 5050 . . . . 5 ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶𝑊) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹⟩ ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐸⟩ ∧ 𝐶 = 𝐹)))
64, 5mpan 708 . . . 4 (𝐶𝑊 → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹⟩ ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐸⟩ ∧ 𝐶 = 𝐹)))
7 opthg 5050 . . . . . 6 ((𝐴𝑈𝐵𝑉) → (⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐸⟩ ↔ (𝐴 = 𝐷𝐵 = 𝐸)))
87anbi1d 743 . . . . 5 ((𝐴𝑈𝐵𝑉) → ((⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐸⟩ ∧ 𝐶 = 𝐹) ↔ ((𝐴 = 𝐷𝐵 = 𝐸) ∧ 𝐶 = 𝐹)))
9 df-3an 1074 . . . . 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 739 . . 3 (((𝐴𝑈𝐵𝑉) ∧ 𝐶𝑊) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹⟩ ↔ (𝐴 = 𝐷𝐵 = 𝐸𝐶 = 𝐹)))
12113impa 1100 . 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 383   ∧ w3a 1072   = wceq 1596   ∈ wcel 2103  Vcvv 3304  ⟨cop 4291  ⟨cotp 4293 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-ot 4294 This theorem is referenced by:  otsndisj  5083  otiunsndisj  5084  otiunsndisjX  41723
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