ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nbn2 GIF version

Theorem nbn2 687
Description: The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by Juha Arpiainen, 19-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2015.)
Assertion
Ref Expression
nbn2 𝜑 → (¬ 𝜓 ↔ (𝜑𝜓)))

Proof of Theorem nbn2
StepHypRef Expression
1 pm5.21im 686 . 2 𝜑 → (¬ 𝜓 → (𝜑𝜓)))
2 bi2 129 . . 3 ((𝜑𝜓) → (𝜓𝜑))
3 mtt 675 . . 3 𝜑 → (¬ 𝜓 ↔ (𝜓𝜑)))
42, 3syl5ibr 155 . 2 𝜑 → ((𝜑𝜓) → ¬ 𝜓))
51, 4impbid 128 1 𝜑 → (¬ 𝜓 ↔ (𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  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 604  ax-in2 605
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  bibif  688  pm5.18dc  869  biassdc  1374
  Copyright terms: Public domain W3C validator