Theorem con1dc 858

Description: Contraposition for a decidable proposition. Based on theorem *2.15 of [WhiteheadRussell] p. 102. (Contributed by Jim Kingdon, 29-Mar-2018.)

Assertion

Ref	Expression
con1dc	⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)))

Proof of Theorem con1dc

Step	Hyp	Ref	Expression
1		notnot 630	. . 3 ⊢ (𝜓 → ¬ ¬ 𝜓)
2	1	imim2i 12	. 2 ⊢ ((¬ 𝜑 → 𝜓) → (¬ 𝜑 → ¬ ¬ 𝜓))
3		condc 855	. 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 → ¬ ¬ 𝜓) → (¬ 𝜓 → 𝜑)))
4	2, 3	syl5 32	1 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837

This theorem is referenced by:  impidc  860  simplimdc  862  con1biimdc  875  con1bdc  880  pm3.13dc  962  necon1aidc  2428  necon1bidc  2429  necon1addc  2453  necon1bddc  2454  exmidapne  7379  bitsinv1lem  12316  phiprmpw  12588  fldivp1  12715  prmpwdvds  12722