Theorem imordc 887

Description: Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 711, 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 862	. . 3 |- (DECID ph -> ( ph <-> -. -. ph ) )
2	1	imbi1d 230	. 2 |- (DECID ph -> ( ( ph -> ps ) <-> ( -. -. ph -> ps ) ) )
3		dcn 832	. . 3 |- (DECID ph -> DECID -. ph )
4		dfordc 882	. . 3 |- (DECID -. ph -> ( ( -. ph \/ ps ) <-> ( -. -. ph -> ps ) ) )
5	3, 4	syl 14	. 2 |- (DECID ph -> ( ( -. ph \/ ps ) <-> ( -. -. ph -> ps ) ) )
6	2, 5	bitr4d 190	1 |- (DECID ph -> ( ( ph -> ps ) <-> ( -. ph \/ ps ) ) )

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:  pm4.62dc  888  pm2.26dc  897  nf4dc  1658  algcvgblem  11981  divgcdodd  12075