Theorem con1dc 851

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 624	. . 3 ⊢ (𝜓 → ¬ ¬ 𝜓)
2	1	imim2i 12	. 2 ⊢ ((¬ 𝜑 → 𝜓) → (¬ 𝜑 → ¬ ¬ 𝜓))
3		condc 848	. 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 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:  impidc  853  simplimdc  855  con1biimdc  868  con1bdc  873  pm3.13dc  954  necon1aidc  2391  necon1bidc  2392  necon1addc  2416  necon1bddc  2417  phiprmpw  12163  fldivp1  12287  prmpwdvds  12294