Theorem pm5.12dc 900

Description: Excluded middle with antecedents for a decidable consequent. Based on theorem *5.12 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 30-Mar-2018.)

Assertion

Ref	Expression
pm5.12dc	⊢ (DECID 𝜓 → ((𝜑 → 𝜓) ∨ (𝜑 → ¬ 𝜓)))

Proof of Theorem pm5.12dc

Step	Hyp	Ref	Expression
1		df-dc 825	. 2 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
2		ax-1 6	. . 3 ⊢ (𝜓 → (𝜑 → 𝜓))
3		ax-1 6	. . 3 ⊢ (¬ 𝜓 → (𝜑 → ¬ 𝜓))
4	2, 3	orim12i 749	. 2 ⊢ ((𝜓 ∨ ¬ 𝜓) → ((𝜑 → 𝜓) ∨ (𝜑 → ¬ 𝜓)))
5	1, 4	sylbi 120	1 ⊢ (DECID 𝜓 → ((𝜑 → 𝜓) ∨ (𝜑 → ¬ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 698  DECID wdc 824

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-io 699

This theorem depends on definitions:  df-bi 116  df-dc 825

This theorem is referenced by: (None)