Theorem con1dc 841

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 618	. . 3 ⊢ (𝜓 → ¬ ¬ 𝜓)
2	1	imim2i 12	. 2 ⊢ ((¬ 𝜑 → 𝜓) → (¬ 𝜑 → ¬ ¬ 𝜓))
3		condc 838	. 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 819

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 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820

This theorem is referenced by:  impidc  843  simplimdc  845  con1biimdc  858  con1bdc  863  pm3.13dc  943  necon1aidc  2359  necon1bidc  2360  necon1addc  2384  necon1bddc  2385  phiprmpw  11898