Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ne3anior GIF version

Theorem ne3anior 2396
 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 2309 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 df-ne 2309 . . 3 (𝐶𝐷 ↔ ¬ 𝐶 = 𝐷)
3 df-ne 2309 . . 3 (𝐸𝐹 ↔ ¬ 𝐸 = 𝐹)
41, 2, 33anbi123i 1170 . 2 ((𝐴𝐵𝐶𝐷𝐸𝐹) ↔ (¬ 𝐴 = 𝐵 ∧ ¬ 𝐶 = 𝐷 ∧ ¬ 𝐸 = 𝐹))
5 3ioran 977 . 2 (¬ (𝐴 = 𝐵𝐶 = 𝐷𝐸 = 𝐹) ↔ (¬ 𝐴 = 𝐵 ∧ ¬ 𝐶 = 𝐷 ∧ ¬ 𝐸 = 𝐹))
64, 5bitr4i 186 1 ((𝐴𝐵𝐶𝐷𝐸𝐹) ↔ ¬ (𝐴 = 𝐵𝐶 = 𝐷𝐸 = 𝐹))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ↔ wb 104   ∨ w3o 961   ∧ w3a 962   = wceq 1331   ≠ wne 2308 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698 This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-ne 2309 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator