Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ne3anior Structured version   Visualization version   GIF version

Theorem ne3anior 3035
 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 1096 . 2 ((𝐴𝐵𝐶𝐷𝐸𝐹) ↔ ¬ (¬ 𝐴𝐵 ∨ ¬ 𝐶𝐷 ∨ ¬ 𝐸𝐹))
2 nne 2946 . . 3 𝐴𝐵𝐴 = 𝐵)
3 nne 2946 . . 3 𝐶𝐷𝐶 = 𝐷)
4 nne 2946 . . 3 𝐸𝐹𝐸 = 𝐹)
52, 3, 43orbi123i 1158 . 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 1069   ∧ w3a 1070   = wceq 1630   ≠ wne 2942 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-an 383  df-or 827  df-3or 1071  df-3an 1072  df-ne 2943 This theorem is referenced by:  eldiftp  4363
 Copyright terms: Public domain W3C validator