Mathbox for Anthony Hart < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df3nandALT2 Structured version   Visualization version   GIF version

Theorem df3nandALT2 32522
 Description: The double nand expressed in terms of negation and and not. (Contributed by Anthony Hart, 13-Sep-2011.)
Assertion
Ref Expression
df3nandALT2 ((𝜑𝜓𝜒) ↔ ¬ (𝜑𝜓𝜒))

Proof of Theorem df3nandALT2
StepHypRef Expression
1 df-3nand 32520 . 2 ((𝜑𝜓𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒)))
2 imnan 437 . . 3 ((𝜓 → ¬ 𝜒) ↔ ¬ (𝜓𝜒))
32imbi2i 325 . 2 ((𝜑 → (𝜓 → ¬ 𝜒)) ↔ (𝜑 → ¬ (𝜓𝜒)))
4 imnan 437 . . 3 ((𝜑 → ¬ (𝜓𝜒)) ↔ ¬ (𝜑 ∧ (𝜓𝜒)))
5 3anass 1059 . . 3 ((𝜑𝜓𝜒) ↔ (𝜑 ∧ (𝜓𝜒)))
64, 5xchbinxr 324 . 2 ((𝜑 → ¬ (𝜓𝜒)) ↔ ¬ (𝜑𝜓𝜒))
71, 3, 63bitri 286 1 ((𝜑𝜓𝜒) ↔ ¬ (𝜑𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   ⊼ w3nand 32519 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 385  df-3an 1056  df-3nand 32520 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator