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

Theorem notbi 638
Description: Equivalence property for negation. Closed form. (Contributed by BJ, 27-Jan-2020.)
Assertion
Ref Expression
notbi ((𝜑𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))

Proof of Theorem notbi
StepHypRef Expression
1 bi2 129 . . 3 ((𝜑𝜓) → (𝜓𝜑))
21con3d 603 . 2 ((𝜑𝜓) → (¬ 𝜑 → ¬ 𝜓))
3 bi1 117 . . 3 ((𝜑𝜓) → (𝜑𝜓))
43con3d 603 . 2 ((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
52, 4impbid 128 1 ((𝜑𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
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 586  ax-in2 587
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  notbid  639  notbii  640  ifbi  3460
  Copyright terms: Public domain W3C validator