Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df3an2 Structured version   Visualization version   GIF version

Theorem df3an2 40401
 Description: Express triple-and in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 25-Jul-2020.)
Assertion
Ref Expression
df3an2 ((𝜑𝜓𝜒) ↔ ¬ (𝜑 → (𝜓 → ¬ 𝜒)))

Proof of Theorem df3an2
StepHypRef Expression
1 df-3an 1086 . 2 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
2 df-an 400 . . 3 (((𝜑𝜓) ∧ 𝜒) ↔ ¬ ((𝜑𝜓) → ¬ 𝜒))
3 impexp 454 . . 3 (((𝜑𝜓) → ¬ 𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒)))
42, 3xchbinx 337 . 2 (((𝜑𝜓) ∧ 𝜒) ↔ ¬ (𝜑 → (𝜓 → ¬ 𝜒)))
51, 4bitri 278 1 ((𝜑𝜓𝜒) ↔ ¬ (𝜑 → (𝜓 → ¬ 𝜒)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084 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 210  df-an 400  df-3an 1086 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator