Theorem pm5.11dc 815

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 784	. 2 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 → 𝜓)))
2		pm2.5dc 763	. . 3 ⊢ (DECID 𝜑 → (¬ (𝜑 → 𝜓) → (¬ 𝜑 → 𝜓)))
3		pm2.54dc 790	. . 3 ⊢ (DECID (𝜑 → 𝜓) → ((¬ (𝜑 → 𝜓) → (¬ 𝜑 → 𝜓)) → ((𝜑 → 𝜓) ∨ (¬ 𝜑 → 𝜓))))
4	2, 3	syl5com 26	. 2 ⊢ (DECID 𝜑 → (DECID (𝜑 → 𝜓) → ((𝜑 → 𝜓) ∨ (¬ 𝜑 → 𝜓))))
5	1, 4	syld 40	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 → 𝜓) ∨ (¬ 𝜑 → 𝜓))))

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-in1 544  ax-in2 545  ax-io 630

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

This theorem is referenced by: (None)