Theorem pm5.15dc 1320

Description: A decidable proposition is equivalent to a decidable proposition or its negation. Based on theorem *5.15 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.)

Assertion

Ref	Expression
pm5.15dc	|- (DECID ph -> (DECID ps -> ( ( ph <-> ps ) \/ ( ph <-> -. ps ) ) ) )

Proof of Theorem pm5.15dc

Step	Hyp	Ref	Expression
1		xor3dc 1318	. . . . 5 |- (DECID ph -> (DECID ps -> ( -. ( ph <-> ps ) <-> ( ph <-> -. ps ) ) ) )
2	1	imp 122	. . . 4 |- ( (DECID ph /\ DECID ps ) -> ( -. ( ph <-> ps ) <-> ( ph <-> -. ps ) ) )
3	2	biimpd 142	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( -. ( ph <-> ps ) -> ( ph <-> -. ps ) ) )
4		dcbi 877	. . . . 5 |- (DECID ph -> (DECID ps -> DECID ( ph <-> ps ) ) )
5	4	imp 122	. . . 4 |- ( (DECID ph /\ DECID ps ) -> DECID ( ph <-> ps ) )
6		dfordc 824	. . . 4 |- (DECID ( ph <-> ps ) -> ( ( ( ph <-> ps ) \/ ( ph <-> -. ps ) ) <-> ( -. ( ph <-> ps ) -> ( ph <-> -. ps ) ) ) )
7	5, 6	syl 14	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( ( ph <-> ps ) \/ ( ph <-> -. ps ) ) <-> ( -. ( ph <-> ps ) -> ( ph <-> -. ps ) ) ) )
8	3, 7	mpbird 165	. 2 |- ( (DECID ph /\ DECID ps ) -> ( ( ph <-> ps ) \/ ( ph <-> -. ps ) ) )
9	8	ex 113	1 |- (DECID ph -> (DECID ps -> ( ( ph <-> ps ) \/ ( ph <-> -. ps ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 661  DECID wdc 775

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by: (None)