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Theorem ne3anior 2343
Description: A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.) (Proof rewritten by Jim Kingdon, 19-May-2018.)
Assertion
Ref Expression
ne3anior ((𝐴𝐵𝐶𝐷𝐸𝐹) ↔ ¬ (𝐴 = 𝐵𝐶 = 𝐷𝐸 = 𝐹))

Proof of Theorem ne3anior
StepHypRef Expression
1 df-ne 2256 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 df-ne 2256 . . 3 (𝐶𝐷 ↔ ¬ 𝐶 = 𝐷)
3 df-ne 2256 . . 3 (𝐸𝐹 ↔ ¬ 𝐸 = 𝐹)
41, 2, 33anbi123i 1132 . 2 ((𝐴𝐵𝐶𝐷𝐸𝐹) ↔ (¬ 𝐴 = 𝐵 ∧ ¬ 𝐶 = 𝐷 ∧ ¬ 𝐸 = 𝐹))
5 3ioran 939 . 2 (¬ (𝐴 = 𝐵𝐶 = 𝐷𝐸 = 𝐹) ↔ (¬ 𝐴 = 𝐵 ∧ ¬ 𝐶 = 𝐷 ∧ ¬ 𝐸 = 𝐹))
64, 5bitr4i 185 1 ((𝐴𝐵𝐶𝐷𝐸𝐹) ↔ ¬ (𝐴 = 𝐵𝐶 = 𝐷𝐸 = 𝐹))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 103  w3o 923  w3a 924   = wceq 1289  wne 2255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-ne 2256
This theorem is referenced by: (None)
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