Theorem otth 5489

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 4635	. . 3 ⊢ ⟨𝐴, 𝐵, 𝑅⟩ = ⟨⟨𝐴, 𝐵⟩, 𝑅⟩
2		df-ot 4635	. . 3 ⊢ ⟨𝐶, 𝐷, 𝑆⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩
3	1, 2	eqeq12i 2755	. 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 5488	. 2 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆))
8	3, 7	bitri 275	1 ⊢ (⟨𝐴, 𝐵, 𝑅⟩ = ⟨𝐶, 𝐷, 𝑆⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆))

Syntax hints:   ↔ wb 206   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  Vcvv 3480  ⟨cop 4632  ⟨cotp 4634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-ot 4635

This theorem is referenced by:  otthne  5491  euotd  5518  mthmpps  35587  ackval40  48614