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Theorem imordc 830
Description: Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 831, holds for all propositions. (Contributed by Jim Kingdon, 20-Apr-2018.)
Assertion
Ref Expression
imordc (DECID 𝜑 → ((𝜑𝜓) ↔ (¬ 𝜑𝜓)))

Proof of Theorem imordc
StepHypRef Expression
1 notnotbdc 800 . . 3 (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
21imbi1d 229 . 2 (DECID 𝜑 → ((𝜑𝜓) ↔ (¬ ¬ 𝜑𝜓)))
3 dcn 780 . . 3 (DECID 𝜑DECID ¬ 𝜑)
4 dfordc 825 . . 3 (DECID ¬ 𝜑 → ((¬ 𝜑𝜓) ↔ (¬ ¬ 𝜑𝜓)))
53, 4syl 14 . 2 (DECID 𝜑 → ((¬ 𝜑𝜓) ↔ (¬ ¬ 𝜑𝜓)))
62, 5bitr4d 189 1 (DECID 𝜑 → ((𝜑𝜓) ↔ (¬ 𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  wo 662  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:  pm4.62dc  832  pm2.26dc  847  nf4dc  1601  algcvgblem  10809  divgcdodd  10900
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