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

Theorem nbn 361
Description: The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
Hypothesis
Ref Expression
nbn.1 ¬ 𝜑
Assertion
Ref Expression
nbn 𝜓 ↔ (𝜓𝜑))

Proof of Theorem nbn
StepHypRef Expression
1 nbn.1 . . 3 ¬ 𝜑
2 bibif 360 . . 3 𝜑 → ((𝜓𝜑) ↔ ¬ 𝜓))
31, 2ax-mp 5 . 2 ((𝜓𝜑) ↔ ¬ 𝜓)
43bicomi 214 1 𝜓 ↔ (𝜓𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196
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
This theorem is referenced by:  nbn3  362  nbfal  1535  eq0f  3958  n0fOLD  3961  disj  4050  axnulALT  4820  dm0rn0  5374  reldm0  5375  isarchi  29864
  Copyright terms: Public domain W3C validator