Theorem impidc 853

Description: An importation inference for a decidable consequent. (Contributed by Jim Kingdon, 30-Apr-2018.)

Hypothesis

Ref	Expression
impidc.1	⊢ (DECID 𝜒 → (𝜑 → (𝜓 → 𝜒)))

Assertion

Ref	Expression
impidc	⊢ (DECID 𝜒 → (¬ (𝜑 → ¬ 𝜓) → 𝜒))

Proof of Theorem impidc

Step	Hyp	Ref	Expression
1		impidc.1	. . . . . 6 ⊢ (DECID 𝜒 → (𝜑 → (𝜓 → 𝜒)))
2	1	imp 123	. . . . 5 ⊢ ((DECID 𝜒 ∧ 𝜑) → (𝜓 → 𝜒))
3	2	con3d 626	. . . 4 ⊢ ((DECID 𝜒 ∧ 𝜑) → (¬ 𝜒 → ¬ 𝜓))
4	3	ex 114	. . 3 ⊢ (DECID 𝜒 → (𝜑 → (¬ 𝜒 → ¬ 𝜓)))
5	4	com23 78	. 2 ⊢ (DECID 𝜒 → (¬ 𝜒 → (𝜑 → ¬ 𝜓)))
6		con1dc 851	. 2 ⊢ (DECID 𝜒 → ((¬ 𝜒 → (𝜑 → ¬ 𝜓)) → (¬ (𝜑 → ¬ 𝜓) → 𝜒)))
7	5, 6	mpd 13	1 ⊢ (DECID 𝜒 → (¬ (𝜑 → ¬ 𝜓) → 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  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-in1 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830

This theorem is referenced by:  simprimdc  854