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Theorem con34bdc 799
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 604 . 2 ((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
2 condc 783 . 2 (DECID 𝜓 → ((¬ 𝜓 → ¬ 𝜑) → (𝜑𝜓)))
31, 2impbid2 141 1 (DECID 𝜓 → ((𝜑𝜓) ↔ (¬ 𝜓 → ¬ 𝜑)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  DECID wdc 776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by:  pm4.14dc  821  algcvgblem  10575
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