Theorem dfordc 835

Description: Definition of 'or' in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 682, 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		pm2.53 682	. 2 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓))
2		pm2.54dc 834	. 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 670  DECID wdc 786

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671

This theorem depends on definitions:  df-bi 116  df-dc 787

This theorem is referenced by:  imordc  840  pm4.64dc  845  pm5.17dc  854  pm5.6dc  879  pm3.12dc  910  pm5.15dc  1335  19.32dc  1625  r19.32vdc  2538  prime  9002  isprm4  11593  prm2orodd  11600  euclemma  11617  phiprmpw  11690