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Theorem dfordc 892
Description: Definition of disjunction in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 722, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 26-Mar-2018.)
Assertion
Ref Expression
dfordc (DECID 𝜑 → ((𝜑𝜓) ↔ (¬ 𝜑𝜓)))

Proof of Theorem dfordc
StepHypRef Expression
1 pm2.53 722 . 2 ((𝜑𝜓) → (¬ 𝜑𝜓))
2 pm2.54dc 891 . 2 (DECID 𝜑 → ((¬ 𝜑𝜓) → (𝜑𝜓)))
31, 2impbid2 143 1 (DECID 𝜑 → ((𝜑𝜓) ↔ (¬ 𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 105  wo 708  DECID wdc 834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709
This theorem depends on definitions:  df-bi 117  df-dc 835
This theorem is referenced by:  imordc  897  pm4.64dc  900  pm5.17dc  904  pm5.6dc  926  pm3.12dc  958  pm5.15dc  1389  19.32dc  1679  r19.30dc  2624  r19.32vdc  2626  prime  9350  isprm4  12113  prm2orodd  12120  euclemma  12140  phiprmpw  12216
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