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 → ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)))

Proof of Theorem dfordc

Step	Hyp	Ref	Expression
1		pm2.53 722	. 2 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓))
2		pm2.54dc 891	. 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓)))
3	1, 2	impbid2 143	1 ⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)))

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  9354  isprm4  12121  prm2orodd  12128  euclemma  12148  phiprmpw  12224