Theorem imordc 897

Description: Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 721, holds for all propositions. (Contributed by Jim Kingdon, 20-Apr-2018.)

Assertion

Ref	Expression
imordc	|- (DECID ph -> ( ( ph -> ps ) <-> ( -. ph \/ ps ) ) )

Proof of Theorem imordc

Step	Hyp	Ref	Expression
1		notnotbdc 872	. . 3 |- (DECID ph -> ( ph <-> -. -. ph ) )
2	1	imbi1d 231	. 2 |- (DECID ph -> ( ( ph -> ps ) <-> ( -. -. ph -> ps ) ) )
3		dcn 842	. . 3 |- (DECID ph -> DECID -. ph )
4		dfordc 892	. . 3 |- (DECID -. ph -> ( ( -. ph \/ ps ) <-> ( -. -. ph -> ps ) ) )
5	3, 4	syl 14	. 2 |- (DECID ph -> ( ( -. ph \/ ps ) <-> ( -. -. ph -> ps ) ) )
6	2, 5	bitr4d 191	1 |- (DECID ph -> ( ( ph -> ps ) <-> ( -. ph \/ ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   \/ wo 708  DECID wdc 834

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 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117  df-dc 835

This theorem is referenced by:  pm4.62dc  898  pm2.26dc  907  nf4dc  1670  algcvgblem  12051  divgcdodd  12145