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Theorem ne3anior 2916
Description: A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.)
Assertion
Ref Expression
ne3anior ((𝐴𝐵𝐶𝐷𝐸𝐹) ↔ ¬ (𝐴 = 𝐵𝐶 = 𝐷𝐸 = 𝐹))

Proof of Theorem ne3anior
StepHypRef Expression
1 3anor 1075 . 2 ((𝐴𝐵𝐶𝐷𝐸𝐹) ↔ ¬ (¬ 𝐴𝐵 ∨ ¬ 𝐶𝐷 ∨ ¬ 𝐸𝐹))
2 nne 2827 . . 3 𝐴𝐵𝐴 = 𝐵)
3 nne 2827 . . 3 𝐶𝐷𝐶 = 𝐷)
4 nne 2827 . . 3 𝐸𝐹𝐸 = 𝐹)
52, 3, 43orbi123i 1271 . 2 ((¬ 𝐴𝐵 ∨ ¬ 𝐶𝐷 ∨ ¬ 𝐸𝐹) ↔ (𝐴 = 𝐵𝐶 = 𝐷𝐸 = 𝐹))
61, 5xchbinx 323 1 ((𝐴𝐵𝐶𝐷𝐸𝐹) ↔ ¬ (𝐴 = 𝐵𝐶 = 𝐷𝐸 = 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  w3o 1053  w3a 1054   = wceq 1523  wne 2823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-ne 2824
This theorem is referenced by:  eldiftp  4260
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