Theorem pm5.14dc 817

Description: A decidable proposition is implied by or implies other propositions. Based on theorem *5.14 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 30-Mar-2018.)

Assertion

Ref	Expression
pm5.14dc	⊢ (DECID 𝜓 → ((𝜑 → 𝜓) ∨ (𝜓 → 𝜒)))

Proof of Theorem pm5.14dc

Step	Hyp	Ref	Expression
1		df-dc 743	. 2 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
2		ax-1 5	. . 3 ⊢ (𝜓 → (𝜑 → 𝜓))
3		ax-in2 545	. . 3 ⊢ (¬ 𝜓 → (𝜓 → 𝜒))
4	2, 3	orim12i 676	. 2 ⊢ ((𝜓 ∨ ¬ 𝜓) → ((𝜑 → 𝜓) ∨ (𝜓 → 𝜒)))
5	1, 4	sylbi 114	1 ⊢ (DECID 𝜓 → ((𝜑 → 𝜓) ∨ (𝜓 → 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 629  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by:  pm5.13dc  818