Theorem pm2.6dc 759

Description: Case elimination for a decidable proposition. Based on theorem *2.6 of [WhiteheadRussell] p. 107. (Contributed by Jim Kingdon, 25-Mar-2018.)

Assertion

Ref	Expression
pm2.6dc	⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)))

Proof of Theorem pm2.6dc

Step	Hyp	Ref	Expression
1		pm2.1dc 745	. . 3 ⊢ (DECID 𝜑 → (¬ 𝜑 ∨ 𝜑))
2		pm3.44 635	. . 3 ⊢ (((¬ 𝜑 → 𝜓) ∧ (𝜑 → 𝜓)) → ((¬ 𝜑 ∨ 𝜑) → 𝜓))
3	1, 2	syl5com 26	. 2 ⊢ (DECID 𝜑 → (((¬ 𝜑 → 𝜓) ∧ (𝜑 → 𝜓)) → 𝜓))
4	3	expd 245	1 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∨ 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-io 630

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

This theorem is referenced by:  jadc  760  jaddc  761  pm2.61dc  762