Theorem xor2dc 1368

Description: Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 17-Apr-2018.)

Assertion

Ref	Expression
xor2dc	|- (DECID ph -> (DECID ps -> ( -. ( ph <-> ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) ) ) )

Proof of Theorem xor2dc

Step	Hyp	Ref	Expression
1		xor3dc 1365	. . . 4 |- (DECID ph -> (DECID ps -> ( -. ( ph <-> ps ) <-> ( ph <-> -. ps ) ) ) )
2	1	imp 123	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( -. ( ph <-> ps ) <-> ( ph <-> -. ps ) ) )
3		pm5.17dc 889	. . . 4 |- (DECID ps -> ( ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) <-> ( ph <-> -. ps ) ) )
4	3	adantl 275	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) <-> ( ph <-> -. ps ) ) )
5	2, 4	bitr4d 190	. 2 |- ( (DECID ph /\ DECID ps ) -> ( -. ( ph <-> ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) ) )
6	5	ex 114	1 |- (DECID ph -> (DECID ps -> ( -. ( ph <-> ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697  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:  xornbidc  1369