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Theorem otth2 5494
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 = 𝐶𝐵 = 𝐷𝑅 = 𝑆))

Proof of Theorem otth2
StepHypRef Expression
1 otth.1 . . . 4 𝐴 ∈ V
2 otth.2 . . . 4 𝐵 ∈ V
31, 2opth 5487 . . 3 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
43anbi1i 624 . 2 ((⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∧ 𝑅 = 𝑆) ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∧ 𝑅 = 𝑆))
5 opex 5475 . . 3 𝐴, 𝐵⟩ ∈ V
6 otth.3 . . 3 𝑅 ∈ V
75, 6opth 5487 . 2 (⟨⟨𝐴, 𝐵⟩, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩ ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∧ 𝑅 = 𝑆))
8 df-3an 1088 . 2 ((𝐴 = 𝐶𝐵 = 𝐷𝑅 = 𝑆) ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∧ 𝑅 = 𝑆))
94, 7, 83bitr4i 303 1 (⟨⟨𝐴, 𝐵⟩, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷𝑅 = 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638
This theorem is referenced by:  otth  5495  oprabidw  7462  oprabid  7463  eloprabga  7541  eloprabgaOLD  7542
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