Theorem otth 4879

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 4134	. . 3 ⊢ ⟨𝐴, 𝐵, 𝑅⟩ = ⟨⟨𝐴, 𝐵⟩, 𝑅⟩
2		df-ot 4134	. . 3 ⊢ ⟨𝐶, 𝐷, 𝑆⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩
3	1, 2	eqeq12i 2624	. 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 4878	. 2 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆))
8	3, 7	bitri 263	1 ⊢ (⟨𝐴, 𝐵, 𝑅⟩ = ⟨𝐶, 𝐷, 𝑆⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆))

Syntax hints:   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131  ⟨cotp 4133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-ot 4134

This theorem is referenced by:  euotd  4900  2spotiundisj  26589  mthmpps  30733