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Theorem dfordc 802
Description: Definition of 'or' in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 651, 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 pm2.53 651 . 2 ((𝜑𝜓) → (¬ 𝜑𝜓))
2 pm2.54dc 801 . 2 (DECID 𝜑 → ((¬ 𝜑𝜓) → (𝜑𝜓)))
31, 2impbid2 135 1 (DECID 𝜑 → ((𝜑𝜓) ↔ (¬ 𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102  wo 639  DECID wdc 753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754
This theorem is referenced by:  imordc  807  pm4.64dc  812  pm5.17dc  821  pm5.6dc  846  pm3.12dc  876  pm5.15dc  1296  19.32dc  1585  r19.32vdc  2476  prime  8396
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