Theorem pm2.61ddc 856

Description: Deduction eliminating a decidable antecedent. (Contributed by Jim Kingdon, 4-May-2018.)

Hypotheses

Ref	Expression
pm2.61ddc.1	⊢ (𝜑 → (𝜓 → 𝜒))
pm2.61ddc.2	⊢ (𝜑 → (¬ 𝜓 → 𝜒))

Assertion

Ref	Expression
pm2.61ddc	⊢ (DECID 𝜓 → (𝜑 → 𝜒))

Proof of Theorem pm2.61ddc

Step	Hyp	Ref	Expression
1		df-dc 830	. 2 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
2		pm2.61ddc.1	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	com12 30	. . 3 ⊢ (𝜓 → (𝜑 → 𝜒))
4		pm2.61ddc.2	. . . 4 ⊢ (𝜑 → (¬ 𝜓 → 𝜒))
5	4	com12 30	. . 3 ⊢ (¬ 𝜓 → (𝜑 → 𝜒))
6	3, 5	jaoi 711	. 2 ⊢ ((𝜓 ∨ ¬ 𝜓) → (𝜑 → 𝜒))
7	1, 6	sylbi 120	1 ⊢ (DECID 𝜓 → (𝜑 → 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ∨ 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:  bijadc  877