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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 → ((𝜑𝜓) ↔ (¬ 𝜓 → ¬ 𝜑)))

Proof of Theorem con34bdc
StepHypRef Expression
1 con3 631 . 2 ((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
2 condc 838 . 2 (DECID 𝜓 → ((¬ 𝜓 → ¬ 𝜑) → (𝜑𝜓)))
31, 2impbid2 142 1 (DECID 𝜓 → ((𝜑𝜓) ↔ (¬ 𝜓 → ¬ 𝜑)))
Colors of variables: wff set class
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  11737
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