Theorem con34bdc 872

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 643	. 2 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
2		condc 854	. 2 ⊢ (DECID 𝜓 → ((¬ 𝜓 → ¬ 𝜑) → (𝜑 → 𝜓)))
3	1, 2	impbid2 143	1 ⊢ (DECID 𝜓 → ((𝜑 → 𝜓) ↔ (¬ 𝜓 → ¬ 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105  DECID wdc 835

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 710

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836

This theorem is referenced by:  pm4.14dc  891  algcvgblem  12242