Theorem otth 4332

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 3677	. . 3 ⊢ ⟨𝐴, 𝐵, 𝑅⟩ = ⟨⟨𝐴, 𝐵⟩, 𝑅⟩
2		df-ot 3677	. . 3 ⊢ ⟨𝐶, 𝐷, 𝑆⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩
3	1, 2	eqeq12i 2243	. 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 4331	. 2 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆))
8	3, 7	bitri 184	1 ⊢ (⟨𝐴, 𝐵, 𝑅⟩ = ⟨𝐶, 𝐷, 𝑆⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆))

Syntax hints:   ↔ wb 105   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  Vcvv 2800  ⟨cop 3670  ⟨cotp 3671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-ot 3677

This theorem is referenced by:  euotd  4345