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

Theorem nan 636
Description: Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)
Assertion
Ref Expression
nan ((𝜑 → ¬ (𝜓𝜒)) ↔ ((𝜑𝜓) → ¬ 𝜒))

Proof of Theorem nan
StepHypRef Expression
1 impexp 254 . 2 (((𝜑𝜓) → ¬ 𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒)))
2 imnan 634 . . 3 ((𝜓 → ¬ 𝜒) ↔ ¬ (𝜓𝜒))
32imbi2i 219 . 2 ((𝜑 → (𝜓 → ¬ 𝜒)) ↔ (𝜑 → ¬ (𝜓𝜒)))
41, 3bitr2i 178 1 ((𝜑 → ¬ (𝜓𝜒)) ↔ ((𝜑𝜓) → ¬ 𝜒))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  pm4.15  800
  Copyright terms: Public domain W3C validator