Theorem con34bdc 765

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 571	. 2 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
2		condc 749	. 2 ⊢ (DECID 𝜓 → ((¬ 𝜓 → ¬ 𝜑) → (𝜑 → 𝜓)))
3	1, 2	impbid2 131	1 ⊢ (DECID 𝜓 → ((𝜑 → 𝜓) ↔ (¬ 𝜓 → ¬ 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by:  pm4.14dc  787  algcvgblem  9888