Theorem pm2.54dc 789

Description: Deriving disjunction from implication for a decidable proposition. Based on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, pm2.53 640, 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 745	. 2 |- (DECID ph -> DECID -. ph)
2		notnot2dc 750	. . . . 5 |- (DECID ph -> (-. -. ph -> ph))
3		orc 632	. . . . 5 |- (ph -> (ph \/ ps))
4	2, 3	syl6 29	. . . 4 |- (DECID ph -> (-. -. ph -> (ph \/ ps)))
5	4	a1d 22	. . 3 |- (DECID ph -> (DECID -. ph -> (-. -. ph -> (ph \/ ps))))
6		olc 631	. . . 4 |- (ps -> (ph \/ ps))
7	6	a1i 9	. . 3 |- (DECID ph -> (ps -> (ph \/ ps)))
8	5, 7	jaddc 760	. 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 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:  dfordc  790  pm2.68dc  792  pm4.79dc  808  pm5.11dc  814