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

Theorem nanimn 1480
Description: Alternative denial in terms of our primitive connectives (implication and negation). (Contributed by WL, 26-Jun-2020.)
Assertion
Ref Expression
nanimn ((𝜑𝜓) ↔ (𝜑 → ¬ 𝜓))

Proof of Theorem nanimn
StepHypRef Expression
1 df-nan 1478 . 2 ((𝜑𝜓) ↔ ¬ (𝜑𝜓))
2 imnan 400 . 2 ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))
31, 2bitr4i 279 1 ((𝜑𝜓) ↔ (𝜑 → ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wnan 1477
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 208  df-an 397  df-nan 1478
This theorem is referenced by:  nancom  1482  nannan  1483  nannot  1485  nanbi1  1487
  Copyright terms: Public domain W3C validator