 Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  pm5.6dc GIF version

Theorem pm5.6dc 871
 Description: Conjunction in antecedent versus disjunction in consequent, for a decidable proposition. Theorem *5.6 of [WhiteheadRussell] p. 125, with decidability condition added. The reverse implication holds for all propositions (see pm5.6r 872). (Contributed by Jim Kingdon, 2-Apr-2018.)
Assertion
Ref Expression
pm5.6dc (DECID 𝜓 → (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))))

Proof of Theorem pm5.6dc
StepHypRef Expression
1 dfordc 827 . . 3 (DECID 𝜓 → ((𝜓𝜒) ↔ (¬ 𝜓𝜒)))
21imbi2d 228 . 2 (DECID 𝜓 → ((𝜑 → (𝜓𝜒)) ↔ (𝜑 → (¬ 𝜓𝜒))))
3 impexp 259 . 2 (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (¬ 𝜓𝜒)))
42, 3syl6rbbr 197 1 (DECID 𝜓 → (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 662  DECID wdc 778 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663 This theorem depends on definitions:  df-bi 115  df-dc 779 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator