Theorem jadc 793

Description: Inference forming an implication from the antecedents of two premises, where a decidable antecedent is negated. (Contributed by Jim Kingdon, 25-Mar-2018.)

Hypotheses

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

Assertion

Ref	Expression
jadc	⊢ (DECID 𝜑 → ((𝜑 → 𝜓) → 𝜒))

Proof of Theorem jadc

Step	Hyp	Ref	Expression
1		jadc.2	. . 3 ⊢ (𝜓 → 𝜒)
2	1	imim2i 12	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜒))
3		jadc.1	. . 3 ⊢ (DECID 𝜑 → (¬ 𝜑 → 𝜒))
4		pm2.6dc 792	. . 3 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜒) → ((𝜑 → 𝜒) → 𝜒)))
5	3, 4	mpd 13	. 2 ⊢ (DECID 𝜑 → ((𝜑 → 𝜒) → 𝜒))
6	2, 5	syl5 32	1 ⊢ (DECID 𝜑 → ((𝜑 → 𝜓) → 𝜒))

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:  pm5.71dc  902