ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pm2.25dc GIF version

Theorem pm2.25dc 893
Description: Elimination of disjunction based on a disjunction, for a decidable proposition. Based on theorem *2.25 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
Assertion
Ref Expression
pm2.25dc (DECID 𝜑 → (𝜑 ∨ ((𝜑𝜓) → 𝜓)))

Proof of Theorem pm2.25dc
StepHypRef Expression
1 df-dc 835 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 orel1 725 . . 3 𝜑 → ((𝜑𝜓) → 𝜓))
32orim2i 761 . 2 ((𝜑 ∨ ¬ 𝜑) → (𝜑 ∨ ((𝜑𝜓) → 𝜓)))
41, 3sylbi 121 1 (DECID 𝜑 → (𝜑 ∨ ((𝜑𝜓) → 𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 708  DECID wdc 834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 615  ax-io 709
This theorem depends on definitions:  df-bi 117  df-dc 835
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator