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Theorem ne3anior 2293
 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 2206 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 df-ne 2206 . . 3 (𝐶𝐷 ↔ ¬ 𝐶 = 𝐷)
3 df-ne 2206 . . 3 (𝐸𝐹 ↔ ¬ 𝐸 = 𝐹)
41, 2, 33anbi123i 1093 . 2 ((𝐴𝐵𝐶𝐷𝐸𝐹) ↔ (¬ 𝐴 = 𝐵 ∧ ¬ 𝐶 = 𝐷 ∧ ¬ 𝐸 = 𝐹))
5 3ioran 900 . 2 (¬ (𝐴 = 𝐵𝐶 = 𝐷𝐸 = 𝐹) ↔ (¬ 𝐴 = 𝐵 ∧ ¬ 𝐶 = 𝐷 ∧ ¬ 𝐸 = 𝐹))
64, 5bitr4i 176 1 ((𝐴𝐵𝐶𝐷𝐸𝐹) ↔ ¬ (𝐴 = 𝐵𝐶 = 𝐷𝐸 = 𝐹))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ↔ wb 98   ∨ w3o 884   ∧ w3a 885   = wceq 1243   ≠ wne 2204 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630 This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-ne 2206 This theorem is referenced by: (None)
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