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Theorem otth2 5352
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 5345 . . 3 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
43anbi1i 626 . 2 ((⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∧ 𝑅 = 𝑆) ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∧ 𝑅 = 𝑆))
5 opex 5333 . . 3 𝐴, 𝐵⟩ ∈ V
6 otth.3 . . 3 𝑅 ∈ V
75, 6opth 5345 . 2 (⟨⟨𝐴, 𝐵⟩, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩ ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∧ 𝑅 = 𝑆))
8 df-3an 1086 . 2 ((𝐴 = 𝐶𝐵 = 𝐷𝑅 = 𝑆) ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∧ 𝑅 = 𝑆))
94, 7, 83bitr4i 306 1 (⟨⟨𝐴, 𝐵⟩, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷𝑅 = 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  Vcvv 3469  cop 4545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546
This theorem is referenced by:  otth  5353  oprabidw  7171  oprabid  7172  eloprabga  7245
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