Theorem jaddc 859

Description: Deduction forming an implication from the antecedents of two premises, where a decidable antecedent is negated. (Contributed by Jim Kingdon, 26-Mar-2018.)

Hypotheses

Ref	Expression
jaddc.1	⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 → 𝜃)))
jaddc.2	⊢ (𝜑 → (𝜒 → 𝜃))

Assertion

Ref	Expression
jaddc	⊢ (𝜑 → (DECID 𝜓 → ((𝜓 → 𝜒) → 𝜃)))

Proof of Theorem jaddc

Step	Hyp	Ref	Expression
1		jaddc.2	. . 3 ⊢ (𝜑 → (𝜒 → 𝜃))
2	1	imim2d 54	. 2 ⊢ (𝜑 → ((𝜓 → 𝜒) → (𝜓 → 𝜃)))
3		jaddc.1	. . 3 ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 → 𝜃)))
4		pm2.6dc 857	. . 3 ⊢ (DECID 𝜓 → ((¬ 𝜓 → 𝜃) → ((𝜓 → 𝜃) → 𝜃)))
5	3, 4	sylcom 28	. 2 ⊢ (𝜑 → (DECID 𝜓 → ((𝜓 → 𝜃) → 𝜃)))
6	2, 5	syl5d 68	1 ⊢ (𝜑 → (DECID 𝜓 → ((𝜓 → 𝜒) → 𝜃)))

Syntax hints:  ¬ wn 3   → wi 4  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:  pm2.54dc  886