Theorem pm4.55dc 940

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 903	. . . . 5 |- (DECID ph -> (DECID ps -> ( ( -. ph /\ ps ) <-> -. ( ph \/ -. ps ) ) ) )
2	1	imp 124	. . . 4 |- ( (DECID ph /\ DECID ps ) -> ( ( -. ph /\ ps ) <-> -. ( ph \/ -. ps ) ) )
3		dcor 937	. . . . . 6 |- (DECID ph -> (DECID -. ps -> DECID ( ph \/ -. ps ) ) )
4		dcn 843	. . . . . 6 |- (DECID ps -> DECID -. ps )
5	3, 4	impel 280	. . . . 5 |- ( (DECID ph /\ DECID ps ) -> DECID ( ph \/ -. ps ) )
6		dcn 843	. . . . . 6 |- (DECID ph -> DECID -. ph )
7		dcan 935	. . . . . 6 |- ( (DECID -. ph /\ DECID ps ) -> DECID ( -. ph /\ ps ) )
8	6, 7	sylan 283	. . . . 5 |- ( (DECID ph /\ DECID ps ) -> DECID ( -. ph /\ ps ) )
9		con2bidc 876	. . . . 5 |- (DECID ( ph \/ -. ps ) -> (DECID ( -. ph /\ ps ) -> ( ( ( ph \/ -. ps ) <-> -. ( -. ph /\ ps ) ) <-> ( ( -. ph /\ ps ) <-> -. ( ph \/ -. ps ) ) ) ) )
10	5, 8, 9	sylc 62	. . . 4 |- ( (DECID ph /\ DECID ps ) -> ( ( ( ph \/ -. ps ) <-> -. ( -. ph /\ ps ) ) <-> ( ( -. ph /\ ps ) <-> -. ( ph \/ -. ps ) ) ) )
11	2, 10	mpbird 167	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( ph \/ -. ps ) <-> -. ( -. ph /\ ps ) ) )
12	11, 11, 11	3bitr2rd 217	. 2 |- ( (DECID ph /\ DECID ps ) -> ( -. ( -. ph /\ ps ) <-> ( ph \/ -. ps ) ) )
13	12	ex 115	1 |- (DECID ph -> (DECID ps -> ( -. ( -. ph /\ ps ) <-> ( ph \/ -. ps ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836

This theorem is referenced by: (None)