Theorem dfordc 877

Description: Definition of disjunction in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 711, 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		pm2.53 711	. 2 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓))
2		pm2.54dc 876	. 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓)))
3	1, 2	impbid2 142	1 ⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   ∨ wo 697  DECID wdc 819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-dc 820

This theorem is referenced by:  imordc  882  pm4.64dc  885  pm5.17dc  889  pm5.6dc  911  pm3.12dc  942  pm5.15dc  1367  19.32dc  1657  r19.32vdc  2578  prime  9143  isprm4  11789  prm2orodd  11796  euclemma  11813  phiprmpw  11887