Theorem otthg 5441

Description: Ordered triple theorem, closed form. (Contributed by Alexander van der Vekens, 10-Mar-2018.)

Assertion

Ref	Expression
otthg	⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹)))

Proof of Theorem otthg

Step	Hyp	Ref	Expression
1		df-ot 4591	. . 3 ⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
2		df-ot 4591	. . 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 5419	. . . . 5 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
5		opthg 5433	. . . . 5 ⊢ ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ 𝑊) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹⟩ ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐸⟩ ∧ 𝐶 = 𝐹)))
6	4, 5	mpan 691	. . . 4 ⊢ (𝐶 ∈ 𝑊 → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹⟩ ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐸⟩ ∧ 𝐶 = 𝐹)))
7		opthg 5433	. . . . . 6 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → (⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐸⟩ ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸)))
8	7	anbi1d 632	. . . . 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)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ⟨cop 4588  ⟨cotp 4590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-ot 4591

This theorem is referenced by:  otsndisj  5475  otiunsndisj  5476  otiunsndisjX  47639