Theorem otthg 5469

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 4600	. . 3 ⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
2		df-ot 4600	. . 3 ⊢ ⟨𝐷, 𝐸, 𝐹⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹⟩
3	1, 2	eqeq12i 2783	. 2 ⊢ (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹⟩)
4		opex 5447	. . . . 5 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
5		opthg 5461	. . . . 5 ⊢ ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ 𝑊) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹⟩ ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐸⟩ ∧ 𝐶 = 𝐹)))
6	4, 5	mpan 703	. . . 4 ⊢ (𝐶 ∈ 𝑊 → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹⟩ ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐸⟩ ∧ 𝐶 = 𝐹)))
7		opthg 5461	. . . . . 6 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → (⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐸⟩ ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸)))
8	7	anbi1d 643	. . . . 5 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → ((⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐸⟩ ∧ 𝐶 = 𝐹) ↔ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐸) ∧ 𝐶 = 𝐹)))
9		df-3an 1105	. . . . 5 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹) ↔ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐸) ∧ 𝐶 = 𝐹))
10	8, 9	bitr4di 292	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → ((⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐸⟩ ∧ 𝐶 = 𝐹) ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹)))
11	6, 10	sylan9bbr 520	. . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝐶 ∈ 𝑊) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹⟩ ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹)))
12	11	3impa 1127	. 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝐷, 𝐸⟩, 𝐹⟩ ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹)))
13	3, 12	bitrid 286	1 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ⟨cop 4597  ⟨cotp 4599

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-ot 4600

This theorem is used by:  otsndisj  5504  otiunsndisj  5505  otiunsndisjX  48076