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Theorem otth 4080
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 = 𝐶𝐵 = 𝐷𝑅 = 𝑆))

Proof of Theorem otth
StepHypRef Expression
1 df-ot 3462 . . 3 𝐴, 𝐵, 𝑅⟩ = ⟨⟨𝐴, 𝐵⟩, 𝑅
2 df-ot 3462 . . 3 𝐶, 𝐷, 𝑆⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆
31, 2eqeq12i 2102 . 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
74, 5, 6otth2 4079 . 2 (⟨⟨𝐴, 𝐵⟩, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷𝑅 = 𝑆))
83, 7bitri 183 1 (⟨𝐴, 𝐵, 𝑅⟩ = ⟨𝐶, 𝐷, 𝑆⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷𝑅 = 𝑆))
Colors of variables: wff set class
Syntax hints:  wb 104  w3a 925   = wceq 1290  wcel 1439  Vcvv 2622  cop 3455  cotp 3456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-ot 3462
This theorem is referenced by:  euotd  4092
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