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Theorem opthneg 5096
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 2931 . 2 (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ ¬ ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
2 opthg 5092 . . . 4 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
32notbid 307 . . 3 ((𝐴𝑉𝐵𝑊) → (¬ ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ ¬ (𝐴 = 𝐶𝐵 = 𝐷)))
4 ianor 510 . . . 4 (¬ (𝐴 = 𝐶𝐵 = 𝐷) ↔ (¬ 𝐴 = 𝐶 ∨ ¬ 𝐵 = 𝐷))
5 df-ne 2931 . . . . 5 (𝐴𝐶 ↔ ¬ 𝐴 = 𝐶)
6 df-ne 2931 . . . . 5 (𝐵𝐷 ↔ ¬ 𝐵 = 𝐷)
75, 6orbi12i 544 . . . 4 ((𝐴𝐶𝐵𝐷) ↔ (¬ 𝐴 = 𝐶 ∨ ¬ 𝐵 = 𝐷))
84, 7bitr4i 267 . . 3 (¬ (𝐴 = 𝐶𝐵 = 𝐷) ↔ (𝐴𝐶𝐵𝐷))
93, 8syl6bb 276 . 2 ((𝐴𝑉𝐵𝑊) → (¬ ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴𝐶𝐵𝐷)))
101, 9syl5bb 272 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴𝐶𝐵𝐷)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1630  wcel 2137  wne 2930  cop 4325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-rab 3057  df-v 3340  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326
This theorem is referenced by:  opthne  5097  zlmodzxznm  42794
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