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Theorem imordc 882
Description: Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 710, 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 857 . . 3 (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
21imbi1d 230 . 2 (DECID 𝜑 → ((𝜑𝜓) ↔ (¬ ¬ 𝜑𝜓)))
3 dcn 827 . . 3 (DECID 𝜑DECID ¬ 𝜑)
4 dfordc 877 . . 3 (DECID ¬ 𝜑 → ((¬ 𝜑𝜓) ↔ (¬ ¬ 𝜑𝜓)))
53, 4syl 14 . 2 (DECID 𝜑 → ((¬ 𝜑𝜓) ↔ (¬ ¬ 𝜑𝜓)))
62, 5bitr4d 190 1 (DECID 𝜑 → ((𝜑𝜓) ↔ (¬ 𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  wo 697  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-dc 820
This theorem is referenced by:  pm4.62dc  883  pm2.26dc  892  nf4dc  1648  algcvgblem  11737  divgcdodd  11828
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