Theorem pm2.25dc 878

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 ∨ ((𝜑 ∨ 𝜓) → 𝜓)))

Proof of Theorem pm2.25dc

Step	Hyp	Ref	Expression
1		df-dc 820	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		orel1 714	. . 3 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) → 𝜓))
3	2	orim2i 750	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → (𝜑 ∨ ((𝜑 ∨ 𝜓) → 𝜓)))
4	1, 3	sylbi 120	1 ⊢ (DECID 𝜑 → (𝜑 ∨ ((𝜑 ∨ 𝜓) → 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 697  DECID wdc 819

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-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-dc 820

This theorem is referenced by: (None)