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Theorem opthneg 5365
Description: Two ordered pairs are not equal iff their first components or their second components are not equal. (Contributed by AV, 13-Dec-2018.)
Assertion
Ref Expression
opthneg ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴𝐶𝐵𝐷)))

Proof of Theorem opthneg
StepHypRef Expression
1 df-ne 2941 . 2 (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ ¬ ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
2 opthg 5361 . . . 4 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
32notbid 321 . . 3 ((𝐴𝑉𝐵𝑊) → (¬ ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ ¬ (𝐴 = 𝐶𝐵 = 𝐷)))
4 ianor 982 . . . 4 (¬ (𝐴 = 𝐶𝐵 = 𝐷) ↔ (¬ 𝐴 = 𝐶 ∨ ¬ 𝐵 = 𝐷))
5 df-ne 2941 . . . . 5 (𝐴𝐶 ↔ ¬ 𝐴 = 𝐶)
6 df-ne 2941 . . . . 5 (𝐵𝐷 ↔ ¬ 𝐵 = 𝐷)
75, 6orbi12i 915 . . . 4 ((𝐴𝐶𝐵𝐷) ↔ (¬ 𝐴 = 𝐶 ∨ ¬ 𝐵 = 𝐷))
84, 7bitr4i 281 . . 3 (¬ (𝐴 = 𝐶𝐵 = 𝐷) ↔ (𝐴𝐶𝐵𝐷))
93, 8bitrdi 290 . 2 ((𝐴𝑉𝐵𝑊) → (¬ ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴𝐶𝐵𝐷)))
101, 9syl5bb 286 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴𝐶𝐵𝐷)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 847   = wceq 1543  wcel 2110  wne 2940  cop 4547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548
This theorem is referenced by:  opthne  5366  addsqnreup  26324  linds2eq  31289  addsval  33863  zlmodzxznm  45511
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