Theorem imordc 902

Description: Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 726, 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 877	. . 3 ⊢ (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
2	1	imbi1d 231	. 2 ⊢ (DECID 𝜑 → ((𝜑 → 𝜓) ↔ (¬ ¬ 𝜑 → 𝜓)))
3		dcn 847	. . 3 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)
4		dfordc 897	. . 3 ⊢ (DECID ¬ 𝜑 → ((¬ 𝜑 ∨ 𝜓) ↔ (¬ ¬ 𝜑 → 𝜓)))
5	3, 4	syl 14	. 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 ∨ 𝜓) ↔ (¬ ¬ 𝜑 → 𝜓)))
6	2, 5	bitr4d 191	1 ⊢ (DECID 𝜑 → ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 713  DECID wdc 839

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 617  ax-in2 618  ax-io 714

This theorem depends on definitions:  df-bi 117  df-dc 840

This theorem is referenced by:  pm4.62dc  903  pm2.26dc  912  dfifp4dc  989  dfifp5dc  990  nf4dc  1716  algcvgblem  12557  divgcdodd  12651