Theorem pm5.11dc 894

Description: A decidable proposition or its negation implies a second proposition. Based on theorem *5.11 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 29-Mar-2018.)

Assertion

Ref	Expression
pm5.11dc	⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 → 𝜓) ∨ (¬ 𝜑 → 𝜓))))

Proof of Theorem pm5.11dc

Step	Hyp	Ref	Expression
1		dcim 826	. 2 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 → 𝜓)))
2		pm2.5dc 852	. . 3 ⊢ (DECID 𝜑 → (¬ (𝜑 → 𝜓) → (¬ 𝜑 → 𝜓)))
3		pm2.54dc 876	. . 3 ⊢ (DECID (𝜑 → 𝜓) → ((¬ (𝜑 → 𝜓) → (¬ 𝜑 → 𝜓)) → ((𝜑 → 𝜓) ∨ (¬ 𝜑 → 𝜓))))
4	2, 3	syl5com 29	. 2 ⊢ (DECID 𝜑 → (DECID (𝜑 → 𝜓) → ((𝜑 → 𝜓) ∨ (¬ 𝜑 → 𝜓))))
5	1, 4	syld 45	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 → 𝜓) ∨ (¬ 𝜑 → 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-in1 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820

This theorem is referenced by: (None)