Theorem imordc 795

Description: Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 796, 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		notnotdc 765	. . 3 ⊢ (DECID φ → (φ ↔ ¬ ¬ φ))
2	1	imbi1d 220	. 2 ⊢ (DECID φ → ((φ → ψ) ↔ (¬ ¬ φ → ψ)))
3		dcn 745	. . 3 ⊢ (DECID φ → DECID ¬ φ)
4		dfordc 790	. . 3 ⊢ (DECID ¬ φ → ((¬ φ ∨ ψ) ↔ (¬ ¬ φ → ψ)))
5	3, 4	syl 14	. 2 ⊢ (DECID φ → ((¬ φ ∨ ψ) ↔ (¬ ¬ φ → ψ)))
6	2, 5	bitr4d 180	1 ⊢ (DECID φ → ((φ → ψ) ↔ (¬ φ ∨ ψ)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   ∨ wo 628  DECID wdc 741

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629

This theorem depends on definitions:  df-bi 110  df-dc 742

This theorem is referenced by:  pm4.62dc  797  pm2.26dc  812  nf4dc  1557