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Theorem imordc 795
Description: Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 796, holds for all propositions. (Contributed by Jim Kingdon, 20-Apr-2018.)
Assertion
Ref Expression
imordc (DECID φ → ((φψ) ↔ (¬ φ ψ)))

Proof of Theorem imordc
StepHypRef Expression
1 notnotdc 765 . . 3 (DECID φ → (φ ↔ ¬ ¬ φ))
21imbi1d 220 . 2 (DECID φ → ((φψ) ↔ (¬ ¬ φψ)))
3 dcn 745 . . 3 (DECID φDECID ¬ φ)
4 dfordc 790 . . 3 (DECID ¬ φ → ((¬ φ ψ) ↔ (¬ ¬ φψ)))
53, 4syl 14 . 2 (DECID φ → ((¬ φ ψ) ↔ (¬ ¬ φψ)))
62, 5bitr4d 180 1 (DECID φ → ((φψ) ↔ (¬ φ ψ)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98   wo 628  DECID wdc 741
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 629
This theorem depends on definitions:  df-bi 110  df-dc 742
This theorem is referenced by:  pm4.62dc  797  pm2.26dc  812  nf4dc  1557
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