Theorem otth2 5459

Description: Ordered triple theorem, with triple expressed with ordered pairs. (Contributed by NM, 1-May-1995.) (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
otth2	⊢ (⟨⟨𝐴, 𝐵⟩, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆))

Proof of Theorem otth2

Step	Hyp	Ref	Expression
1		otth.1	. . . 4 ⊢ 𝐴 ∈ V
2		otth.2	. . . 4 ⊢ 𝐵 ∈ V
3	1, 2	opth 5452	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
4	3	anbi1i 636	. 2 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∧ 𝑅 = 𝑆) ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∧ 𝑅 = 𝑆))
5		opex 5439	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
6		otth.3	. . 3 ⊢ 𝑅 ∈ V
7	5, 6	opth 5452	. 2 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩ ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∧ 𝑅 = 𝑆))
8		df-3an 1105	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆) ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∧ 𝑅 = 𝑆))
9	4, 7, 8	3bitr4i 306	1 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ⟨cop 4840

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

This theorem is used by:  otth  5460  oprabidw  7482  oprabid  7483  eloprabga  7560