Theorem pm2.54dc 876

Description: Deriving disjunction from implication for a decidable proposition. Based on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, pm2.53 711, holds whether the proposition is decidable or not. (Contributed by Jim Kingdon, 26-Mar-2018.)

Assertion

Ref	Expression
pm2.54dc	|- (DECID ph -> ( ( -. ph -> ps ) -> ( ph \/ ps ) ) )

Proof of Theorem pm2.54dc

Step	Hyp	Ref	Expression
1		dcn 827	. 2 |- (DECID ph -> DECID -. ph )
2		notnotrdc 828	. . . . 5 |- (DECID ph -> ( -. -. ph -> ph ) )
3		orc 701	. . . . 5 |- ( ph -> ( ph \/ ps ) )
4	2, 3	syl6 33	. . . 4 |- (DECID ph -> ( -. -. ph -> ( ph \/ ps ) ) )
5	4	a1d 22	. . 3 |- (DECID ph -> (DECID -. ph -> ( -. -. ph -> ( ph \/ ps ) ) ) )
6		olc 700	. . . 4 |- ( ps -> ( ph \/ ps ) )
7	6	a1i 9	. . 3 |- (DECID ph -> ( ps -> ( ph \/ ps ) ) )
8	5, 7	jaddc 849	. 2 |- (DECID ph -> (DECID -. ph -> ( ( -. ph -> ps ) -> ( ph \/ ps ) ) ) )
9	1, 8	mpd 13	1 |- (DECID ph -> ( ( -. ph -> ps ) -> ( ph \/ ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 697  DECID wdc 819

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 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-dc 820

This theorem is referenced by:  dfordc  877  pm2.68dc  879  pm4.79dc  888  pm5.11dc  894