Theorem pm2.85dc 873

Description: Reverse distribution of disjunction over implication, given decidability. Based on theorem *2.85 of [WhiteheadRussell] p. 108. (Contributed by Jim Kingdon, 1-Apr-2018.)

Assertion

Ref	Expression
pm2.85dc	|- (DECID ph -> ( ( ( ph \/ ps ) -> ( ph \/ ch ) ) -> ( ph \/ ( ps -> ch ) ) ) )

Proof of Theorem pm2.85dc

Step	Hyp	Ref	Expression
1		df-dc 803	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		orc 684	. . . 4 |- ( ph -> ( ph \/ ( ps -> ch ) ) )
3	2	a1d 22	. . 3 |- ( ph -> ( ( ( ph \/ ps ) -> ( ph \/ ch ) ) -> ( ph \/ ( ps -> ch ) ) ) )
4		olc 683	. . . . . 6 |- ( ps -> ( ph \/ ps ) )
5	4	imim1i 60	. . . . 5 |- ( ( ( ph \/ ps ) -> ( ph \/ ch ) ) -> ( ps -> ( ph \/ ch ) ) )
6		orel1 697	. . . . 5 |- ( -. ph -> ( ( ph \/ ch ) -> ch ) )
7	5, 6	syl9r 73	. . . 4 |- ( -. ph -> ( ( ( ph \/ ps ) -> ( ph \/ ch ) ) -> ( ps -> ch ) ) )
8		olc 683	. . . 4 |- ( ( ps -> ch ) -> ( ph \/ ( ps -> ch ) ) )
9	7, 8	syl6 33	. . 3 |- ( -. ph -> ( ( ( ph \/ ps ) -> ( ph \/ ch ) ) -> ( ph \/ ( ps -> ch ) ) ) )
10	3, 9	jaoi 688	. 2 |- ( ( ph \/ -. ph ) -> ( ( ( ph \/ ps ) -> ( ph \/ ch ) ) -> ( ph \/ ( ps -> ch ) ) ) )
11	1, 10	sylbi 120	1 |- (DECID ph -> ( ( ( ph \/ ps ) -> ( ph \/ ch ) ) -> ( ph \/ ( ps -> ch ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 680  DECID wdc 802

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 587  ax-io 681

This theorem depends on definitions:  df-bi 116  df-dc 803

This theorem is referenced by:  orimdidc  874