Theorem otth2 4952

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 4945	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
4	3	anbi1i 731	. 2 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∧ 𝑅 = 𝑆) ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∧ 𝑅 = 𝑆))
5		opex 4932	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
6		otth.3	. . 3 ⊢ 𝑅 ∈ V
7	5, 6	opth 4945	. 2 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩ ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∧ 𝑅 = 𝑆))
8		df-3an 1039	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆) ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∧ 𝑅 = 𝑆))
9	4, 7, 8	3bitr4i 292	1 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆))

Syntax hints:   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ⟨cop 4183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184

This theorem is referenced by:  otth  4953  oprabid  6677  eloprabga  6747