Theorem pm2.61ddc 751

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 734	. 2 ⊢ (DECID ψ ↔ (ψ ∨ ¬ ψ))
2		pm2.61ddc.1	. . . 4 ⊢ (φ → (ψ → χ))
3	2	com12 27	. . 3 ⊢ (ψ → (φ → χ))
4		pm2.61ddc.2	. . . 4 ⊢ (φ → (¬ ψ → χ))
5	4	com12 27	. . 3 ⊢ (¬ ψ → (φ → χ))
6	3, 5	jaoi 623	. 2 ⊢ ((ψ ∨ ¬ ψ) → (φ → χ))
7	1, 6	sylbi 114	1 ⊢ (DECID ψ → (φ → χ))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 616  DECID wdc 733

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 617

This theorem depends on definitions:  df-bi 110  df-dc 734

This theorem is referenced by:  bijadc  769