Theorem jaddc 794

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 53	. 2 ⊢ (𝜑 → ((𝜓 → 𝜒) → (𝜓 → 𝜃)))
3		jaddc.1	. . 3 ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 → 𝜃)))
4		pm2.6dc 792	. . 3 ⊢ (DECID 𝜓 → ((¬ 𝜓 → 𝜃) → ((𝜓 → 𝜃) → 𝜃)))
5	3, 4	sylcom 28	. 2 ⊢ (𝜑 → (DECID 𝜓 → ((𝜓 → 𝜃) → 𝜃)))
6	2, 5	syl5d 67	1 ⊢ (𝜑 → (DECID 𝜓 → ((𝜓 → 𝜒) → 𝜃)))

Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 775

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by:  pm2.54dc  823