ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nbn GIF version

Theorem nbn 688
Description: The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by NM, 5-Aug-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 687 . . 3 𝜑 → ((𝜓𝜑) ↔ ¬ 𝜓))
31, 2ax-mp 5 . 2 ((𝜓𝜑) ↔ ¬ 𝜓)
43bicomi 131 1 𝜓 ↔ (𝜓𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  nbn3  689  nbfal  1342  n0rf  3370  eq0  3376  disj  3406  dm0rn0  4751  reldm0  4752
  Copyright terms: Public domain W3C validator