Theorem con34bdc 856

Description: Contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116, but for a decidable proposition. (Contributed by Jim Kingdon, 24-Apr-2018.)

Assertion

Ref	Expression
con34bdc	⊢ (DECID 𝜓 → ((𝜑 → 𝜓) ↔ (¬ 𝜓 → ¬ 𝜑)))

Proof of Theorem con34bdc

Step	Hyp	Ref	Expression
1		con3 631	. 2 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
2		condc 838	. 2 ⊢ (DECID 𝜓 → ((¬ 𝜓 → ¬ 𝜑) → (𝜑 → 𝜓)))
3	1, 2	impbid2 142	1 ⊢ (DECID 𝜓 → ((𝜑 → 𝜓) ↔ (¬ 𝜓 → ¬ 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  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:  pm4.14dc  875  algcvgblem  11730