Theorem pm5.17dc 894

Description: Two ways of stating exclusive-or which are equivalent for a decidable proposition. Based on theorem *5.17 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 16-Apr-2018.)

Assertion

Ref	Expression
pm5.17dc	|- (DECID ps -> ( ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) <-> ( ph <-> -. ps ) ) )

Proof of Theorem pm5.17dc

Step	Hyp	Ref	Expression
1		bicom 139	. 2 |- ( ( ph <-> -. ps ) <-> ( -. ps <-> ph ) )
2		dfbi2 386	. . 3 |- ( ( -. ps <-> ph ) <-> ( ( -. ps -> ph ) /\ ( ph -> -. ps ) ) )
3		orcom 718	. . . . 5 |- ( ( ph \/ ps ) <-> ( ps \/ ph ) )
4		dfordc 882	. . . . 5 |- (DECID ps -> ( ( ps \/ ph ) <-> ( -. ps -> ph ) ) )
5	3, 4	bitr2id 192	. . . 4 |- (DECID ps -> ( ( -. ps -> ph ) <-> ( ph \/ ps ) ) )
6		imnan 680	. . . . 5 |- ( ( ph -> -. ps ) <-> -. ( ph /\ ps ) )
7	6	a1i 9	. . . 4 |- (DECID ps -> ( ( ph -> -. ps ) <-> -. ( ph /\ ps ) ) )
8	5, 7	anbi12d 465	. . 3 |- (DECID ps -> ( ( ( -. ps -> ph ) /\ ( ph -> -. ps ) ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) ) )
9	2, 8	syl5bb 191	. 2 |- (DECID ps -> ( ( -. ps <-> ph ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) ) )
10	1, 9	bitr2id 192	1 |- (DECID ps -> ( ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) <-> ( ph <-> -. ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 824

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 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-dc 825

This theorem is referenced by:  xor2dc  1380