Theorem dfordc 904

Description: Definition of disjunction in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 734, 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		pm2.53 734	. 2 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓))
2		pm2.54dc 903	. 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓)))
3	1, 2	impbid2 143	1 ⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 720  DECID wdc 846

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721

This proof depends on definitions:  df-bi 117  df-dc 847

This theorem is used by:  imordc  909  pm4.64dc  912  pm5.17dc  916  pm5.6dc  938  pm3.12dc  971  pm5.15dc  1438  19.32dc  1731  r19.30dc  2698  r19.32vdc  2700  prime  9745  isprm4  12897  prm2orodd  12904  euclemma  12924  phiprmpw  13000