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Theorem otthne 5487
Description: Contrapositive of the ordered triple theorem. (Contributed by Scott Fenton, 31-Jan-2025.)
Hypotheses
Ref Expression
otthne.1 𝐴 ∈ V
otthne.2 𝐵 ∈ V
otthne.3 𝐶 ∈ V
Assertion
Ref Expression
otthne (⟨𝐴, 𝐵, 𝐶⟩ ≠ ⟨𝐷, 𝐸, 𝐹⟩ ↔ (𝐴𝐷𝐵𝐸𝐶𝐹))

Proof of Theorem otthne
StepHypRef Expression
1 otthne.1 . . . . 5 𝐴 ∈ V
2 otthne.2 . . . . 5 𝐵 ∈ V
3 otthne.3 . . . . 5 𝐶 ∈ V
41, 2, 3otth 5485 . . . 4 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ (𝐴 = 𝐷𝐵 = 𝐸𝐶 = 𝐹))
54notbii 320 . . 3 (¬ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ ¬ (𝐴 = 𝐷𝐵 = 𝐸𝐶 = 𝐹))
6 3ianor 1108 . . 3 (¬ (𝐴 = 𝐷𝐵 = 𝐸𝐶 = 𝐹) ↔ (¬ 𝐴 = 𝐷 ∨ ¬ 𝐵 = 𝐸 ∨ ¬ 𝐶 = 𝐹))
75, 6bitri 275 . 2 (¬ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ (¬ 𝐴 = 𝐷 ∨ ¬ 𝐵 = 𝐸 ∨ ¬ 𝐶 = 𝐹))
8 df-ne 2942 . 2 (⟨𝐴, 𝐵, 𝐶⟩ ≠ ⟨𝐷, 𝐸, 𝐹⟩ ↔ ¬ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩)
9 df-ne 2942 . . 3 (𝐴𝐷 ↔ ¬ 𝐴 = 𝐷)
10 df-ne 2942 . . 3 (𝐵𝐸 ↔ ¬ 𝐵 = 𝐸)
11 df-ne 2942 . . 3 (𝐶𝐹 ↔ ¬ 𝐶 = 𝐹)
129, 10, 113orbi123i 1157 . 2 ((𝐴𝐷𝐵𝐸𝐶𝐹) ↔ (¬ 𝐴 = 𝐷 ∨ ¬ 𝐵 = 𝐸 ∨ ¬ 𝐶 = 𝐹))
137, 8, 123bitr4i 303 1 (⟨𝐴, 𝐵, 𝐶⟩ ≠ ⟨𝐷, 𝐸, 𝐹⟩ ↔ (𝐴𝐷𝐵𝐸𝐶𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  w3o 1087  w3a 1088   = wceq 1542  wcel 2107  wne 2941  Vcvv 3475  cotp 4637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-ot 4638
This theorem is referenced by:  xpord3lem  8135  xpord3pred  8138  xpord3inddlem  8140
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