Theorem otth2 5451

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 5444	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
4	3	anbi1i 633	. 2 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∧ 𝑅 = 𝑆) ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∧ 𝑅 = 𝑆))
5		opex 5431	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
6		otth.3	. . 3 ⊢ 𝑅 ∈ V
7	5, 6	opth 5444	. 2 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩ ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∧ 𝑅 = 𝑆))
8		df-3an 1100	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆) ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∧ 𝑅 = 𝑆))
9	4, 7, 8	3bitr4i 305	1 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆))

Syntax hints:   ↔ wb 208   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  Vcvv 3454  ⟨cop 4588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589

This theorem is referenced by:  otth  5452  oprabidw  7427  oprabid  7428  eloprabga  7505