Theorem pm4.55dc 846

Description: Theorem *4.55 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 2-May-2018.)

Assertion

Ref	Expression
pm4.55dc	|- (DECID ph -> (DECID ps -> ( -. ( -. ph /\ ps ) <-> ( ph \/ -. ps ) ) ) )

Proof of Theorem pm4.55dc

Step	Hyp	Ref	Expression
1		pm4.54dc 805	. . . . 5 |- (DECID ph -> (DECID ps -> ( ( -. ph /\ ps ) <-> -. ( ph \/ -. ps ) ) ) )
2	1	imp 115	. . . 4 |- ( (DECID ph /\ DECID ps ) -> ( ( -. ph /\ ps ) <-> -. ( ph \/ -. ps ) ) )
3		dcn 746	. . . . . . . . 9 |- (DECID ps -> DECID -. ps )
4	3	anim2i 324	. . . . . . . 8 |- ( (DECID ph /\ DECID ps ) -> (DECID ph /\ DECID -. ps ) )
5		dcor 843	. . . . . . . . 9 |- (DECID ph -> (DECID -. ps -> DECID ( ph \/ -. ps ) ) )
6	5	imp 115	. . . . . . . 8 |- ( (DECID ph /\ DECID -. ps ) -> DECID ( ph \/ -. ps ) )
7	4, 6	syl 14	. . . . . . 7 |- ( (DECID ph /\ DECID ps ) -> DECID ( ph \/ -. ps ) )
8		dcn 746	. . . . . . . . 9 |- (DECID ph -> DECID -. ph )
9		dcan 842	. . . . . . . . 9 |- (DECID -. ph -> (DECID ps -> DECID ( -. ph /\ ps ) ) )
10	8, 9	syl 14	. . . . . . . 8 |- (DECID ph -> (DECID ps -> DECID ( -. ph /\ ps ) ) )
11	10	imp 115	. . . . . . 7 |- ( (DECID ph /\ DECID ps ) -> DECID ( -. ph /\ ps ) )
12	7, 11	jca 290	. . . . . 6 |- ( (DECID ph /\ DECID ps ) -> (DECID ( ph \/ -. ps ) /\ DECID ( -. ph /\ ps ) ) )
13		con2bidc 769	. . . . . . 7 |- (DECID ( ph \/ -. ps ) -> (DECID ( -. ph /\ ps ) -> ( ( ( ph \/ -. ps ) <-> -. ( -. ph /\ ps ) ) <-> ( ( -. ph /\ ps ) <-> -. ( ph \/ -. ps ) ) ) ) )
14	13	imp 115	. . . . . 6 |- ( (DECID ( ph \/ -. ps ) /\ DECID ( -. ph /\ ps ) ) -> ( ( ( ph \/ -. ps ) <-> -. ( -. ph /\ ps ) ) <-> ( ( -. ph /\ ps ) <-> -. ( ph \/ -. ps ) ) ) )
15	12, 14	syl 14	. . . . 5 |- ( (DECID ph /\ DECID ps ) -> ( ( ( ph \/ -. ps ) <-> -. ( -. ph /\ ps ) ) <-> ( ( -. ph /\ ps ) <-> -. ( ph \/ -. ps ) ) ) )
16	15	biimprd 147	. . . 4 |- ( (DECID ph /\ DECID ps ) -> ( ( ( -. ph /\ ps ) <-> -. ( ph \/ -. ps ) ) -> ( ( ph \/ -. ps ) <-> -. ( -. ph /\ ps ) ) ) )
17	2, 16	mpd 13	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( ph \/ -. ps ) <-> -. ( -. ph /\ ps ) ) )
18	17	bicomd 129	. 2 |- ( (DECID ph /\ DECID ps ) -> ( -. ( -. ph /\ ps ) <-> ( ph \/ -. ps ) ) )
19	18	ex 108	1 |- (DECID ph -> (DECID ps -> ( -. ( -. ph /\ ps ) <-> ( ph \/ -. ps ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629  DECID wdc 742

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 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by: (None)