Theorem otth 5460

Description: Ordered triple theorem. (Contributed by NM, 25-Sep-2014.) (Revised by Mario Carneiro, 26-Apr-2015.)

Hypotheses

Ref	Expression
otth.1	⊢ 𝐴 ∈ V
otth.2	⊢ 𝐵 ∈ V
otth.3	⊢ 𝑅 ∈ V

Assertion

Ref	Expression
otth	⊢ (⟨𝐴, 𝐵, 𝑅⟩ = ⟨𝐶, 𝐷, 𝑆⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆))

Proof of Theorem otth

Step	Hyp	Ref	Expression
1		df-ot 4864	. . 3 ⊢ ⟨𝐴, 𝐵, 𝑅⟩ = ⟨⟨𝐴, 𝐵⟩, 𝑅⟩
2		df-ot 4864	. . 3 ⊢ ⟨𝐶, 𝐷, 𝑆⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩
3	1, 2	eqeq12i 2781	. 2 ⊢ (⟨𝐴, 𝐵, 𝑅⟩ = ⟨𝐶, 𝐷, 𝑆⟩ ↔ ⟨⟨𝐴, 𝐵⟩, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩)
4		otth.1	. . 3 ⊢ 𝐴 ∈ V
5		otth.2	. . 3 ⊢ 𝐵 ∈ V
6		otth.3	. . 3 ⊢ 𝑅 ∈ V
7	4, 5, 6	otth2 5459	. 2 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆))
8	3, 7	bitri 278	1 ⊢ (⟨𝐴, 𝐵, 𝑅⟩ = ⟨𝐶, 𝐷, 𝑆⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆))

This proof depends on syntax axioms:   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ⟨cop 4840  ⟨cotp 4863

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-ot 4864

This theorem is used by:  otthne  5462  eqvinot  5464  cotsexgw  5470  euotd  5493  mthmpps  36555  ackval40  50136  arweuthinc  50995  arweutermc  50996