Theorem pm4.64dc 890

Description: Theorem *4.64 of [WhiteheadRussell] p. 120, given a decidability condition. The reverse direction, pm2.53 712, holds for all propositions. (Contributed by Jim Kingdon, 2-May-2018.)

Assertion

Ref	Expression
pm4.64dc	⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) ↔ (𝜑 ∨ 𝜓)))

Proof of Theorem pm4.64dc

Step	Hyp	Ref	Expression
1		dfordc 882	. 2 ⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)))
2	1	bicomd 140	1 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) ↔ (𝜑 ∨ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   ∨ wo 698  DECID wdc 824

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 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-dc 825

This theorem is referenced by:  pm4.66dc  891