ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pm4.83dc GIF version

Theorem pm4.83dc 893
Description: Theorem *4.83 of [WhiteheadRussell] p. 122, for decidable propositions. As with other case elimination theorems, like pm2.61dc 796, it only holds for decidable propositions. (Contributed by Jim Kingdon, 12-May-2018.)
Assertion
Ref Expression
pm4.83dc (DECID 𝜑 → (((𝜑𝜓) ∧ (¬ 𝜑𝜓)) ↔ 𝜓))

Proof of Theorem pm4.83dc
StepHypRef Expression
1 df-dc 777 . . 3 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 pm3.44 668 . . . 4 (((𝜑𝜓) ∧ (¬ 𝜑𝜓)) → ((𝜑 ∨ ¬ 𝜑) → 𝜓))
32com12 30 . . 3 ((𝜑 ∨ ¬ 𝜑) → (((𝜑𝜓) ∧ (¬ 𝜑𝜓)) → 𝜓))
41, 3sylbi 119 . 2 (DECID 𝜑 → (((𝜑𝜓) ∧ (¬ 𝜑𝜓)) → 𝜓))
5 ax-1 5 . . 3 (𝜓 → (𝜑𝜓))
6 ax-1 5 . . 3 (𝜓 → (¬ 𝜑𝜓))
75, 6jca 300 . 2 (𝜓 → ((𝜑𝜓) ∧ (¬ 𝜑𝜓)))
84, 7impbid1 140 1 (DECID 𝜑 → (((𝜑𝜓) ∧ (¬ 𝜑𝜓)) ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  DECID wdc 776
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-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator