Theorem xor3dc 1365

Description: Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 12-Apr-2018.)

Assertion

Ref	Expression
xor3dc	|- (DECID ph -> (DECID ps -> ( -. ( ph <-> ps ) <-> ( ph <-> -. ps ) ) ) )

Proof of Theorem xor3dc

Step	Hyp	Ref	Expression
1		dcn 827	. . . . . 6 |- (DECID ps -> DECID -. ps )
2		dcbi 920	. . . . . 6 |- (DECID ph -> (DECID -. ps -> DECID ( ph <-> -. ps ) ) )
3	1, 2	syl5 32	. . . . 5 |- (DECID ph -> (DECID ps -> DECID ( ph <-> -. ps ) ) )
4	3	imp 123	. . . 4 |- ( (DECID ph /\ DECID ps ) -> DECID ( ph <-> -. ps ) )
5		pm5.18dc 868	. . . . . . 7 |- (DECID ph -> (DECID ps -> ( ( ph <-> ps ) <-> -. ( ph <-> -. ps ) ) ) )
6	5	imp 123	. . . . . 6 |- ( (DECID ph /\ DECID ps ) -> ( ( ph <-> ps ) <-> -. ( ph <-> -. ps ) ) )
7	6	a1d 22	. . . . 5 |- ( (DECID ph /\ DECID ps ) -> (DECID ( ph <-> -. ps ) -> ( ( ph <-> ps ) <-> -. ( ph <-> -. ps ) ) ) )
8	7	con2biddc 865	. . . 4 |- ( (DECID ph /\ DECID ps ) -> (DECID ( ph <-> -. ps ) -> ( ( ph <-> -. ps ) <-> -. ( ph <-> ps ) ) ) )
9	4, 8	mpd 13	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( ph <-> -. ps ) <-> -. ( ph <-> ps ) ) )
10	9	bicomd 140	. 2 |- ( (DECID ph /\ DECID ps ) -> ( -. ( ph <-> ps ) <-> ( ph <-> -. ps ) ) )
11	10	ex 114	1 |- (DECID ph -> (DECID ps -> ( -. ( ph <-> ps ) <-> ( ph <-> -. ps ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104  DECID wdc 819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820

This theorem is referenced by:  pm5.15dc  1367  xor2dc  1368  nbbndc  1372