Theorem imordc 829

Description: Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 830, 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 799	. . 3 |- (DECID ph -> ( ph <-> -. -. ph ) )
2	1	imbi1d 229	. 2 |- (DECID ph -> ( ( ph -> ps ) <-> ( -. -. ph -> ps ) ) )
3		dcn 779	. . 3 |- (DECID ph -> DECID -. ph )
4		dfordc 824	. . 3 |- (DECID -. ph -> ( ( -. ph \/ ps ) <-> ( -. -. ph -> ps ) ) )
5	3, 4	syl 14	. 2 |- (DECID ph -> ( ( -. ph \/ ps ) <-> ( -. -. ph -> ps ) ) )
6	2, 5	bitr4d 189	1 |- (DECID ph -> ( ( ph -> ps ) <-> ( -. ph \/ ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   \/ wo 661  DECID wdc 775

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by:  pm4.62dc  831  pm2.26dc  846  nf4dc  1600  algcvgblem  10431  divgcdodd  10522