Theorem pm2.6dc 857

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 832	. . 3 ⊢ (DECID 𝜑 → (¬ 𝜑 ∨ 𝜑))
2		pm3.44 710	. . 3 ⊢ (((¬ 𝜑 → 𝜓) ∧ (𝜑 → 𝜓)) → ((¬ 𝜑 ∨ 𝜑) → 𝜓))
3	1, 2	syl5com 29	. 2 ⊢ (DECID 𝜑 → (((¬ 𝜑 → 𝜓) ∧ (𝜑 → 𝜓)) → 𝜓))
4	3	expd 256	1 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 703  DECID wdc 829

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 704

This theorem depends on definitions:  df-bi 116  df-dc 830

This theorem is referenced by:  jadc  858  jaddc  859  pm2.61dc  860