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

Theorem nannot 1490
Description: Negation in terms of alternative denial. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Revised by Wolf Lammen, 26-Jun-2020.)
Assertion
Ref Expression
nannot 𝜑 ↔ (𝜑𝜑))

Proof of Theorem nannot
StepHypRef Expression
1 nanimn 1485 . 2 ((𝜑𝜑) ↔ (𝜑 → ¬ 𝜑))
2 pm4.8 396 . 2 ((𝜑 → ¬ 𝜑) ↔ ¬ 𝜑)
31, 2bitr2i 279 1 𝜑 ↔ (𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wnan 1482
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-nan 1483
This theorem is referenced by:  nanbi  1491  trunantru  1579  falnanfal  1582  nic-dfneg  1672  andnand1  33776  imnand2  33777
  Copyright terms: Public domain W3C validator