Theorem dfordc 824

Description: Definition of 'or' in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 673, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 26-Mar-2018.)

Assertion

Ref	Expression
dfordc	⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)))

Proof of Theorem dfordc

Step	Hyp	Ref	Expression
1		pm2.53 673	. 2 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓))
2		pm2.54dc 823	. 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓)))
3	1, 2	impbid2 141	1 ⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 103   ∨ wo 661  DECID wdc 775

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 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by:  imordc  829  pm4.64dc  834  pm5.17dc  843  pm5.6dc  868  pm3.12dc  899  pm5.15dc  1320  19.32dc  1609  r19.32vdc  2503  prime  8446  isprm4  10501  prm2orodd  10508  euclemma  10525