Theorem imordc 882

Description: Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 710, holds for all propositions. (Contributed by Jim Kingdon, 20-Apr-2018.)

Assertion

Ref	Expression
imordc	⊢ (DECID 𝜑 → ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓)))

Proof of Theorem imordc

Step	Hyp	Ref	Expression
1		notnotbdc 857	. . 3 ⊢ (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
2	1	imbi1d 230	. 2 ⊢ (DECID 𝜑 → ((𝜑 → 𝜓) ↔ (¬ ¬ 𝜑 → 𝜓)))
3		dcn 827	. . 3 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)
4		dfordc 877	. . 3 ⊢ (DECID ¬ 𝜑 → ((¬ 𝜑 ∨ 𝜓) ↔ (¬ ¬ 𝜑 → 𝜓)))
5	3, 4	syl 14	. 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 ∨ 𝜓) ↔ (¬ ¬ 𝜑 → 𝜓)))
6	2, 5	bitr4d 190	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:  pm4.62dc  883  pm2.26dc  892  nf4dc  1648  algcvgblem  11719  divgcdodd  11810