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Theorem dfordc 827
Description: Definition of 'or' in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 674, 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 pm2.53 674 . 2 ((𝜑𝜓) → (¬ 𝜑𝜓))
2 pm2.54dc 826 . 2 (DECID 𝜑 → ((¬ 𝜑𝜓) → (𝜑𝜓)))
31, 2impbid2 141 1 (DECID 𝜑 → ((𝜑𝜓) ↔ (¬ 𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  wo 662  DECID wdc 778
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 779
This theorem is referenced by:  imordc  832  pm4.64dc  837  pm5.17dc  846  pm5.6dc  871  pm3.12dc  902  pm5.15dc  1323  19.32dc  1612  r19.32vdc  2511  prime  8755  isprm4  10895  prm2orodd  10902  euclemma  10919  phiprmpw  10992
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