Theorem pm5.17dc 843

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 138	. 2 |- ( ( ph <-> -. ps ) <-> ( -. ps <-> ph ) )
2		dfbi2 380	. . 3 |- ( ( -. ps <-> ph ) <-> ( ( -. ps -> ph ) /\ ( ph -> -. ps ) ) )
3		orcom 679	. . . . 5 |- ( ( ph \/ ps ) <-> ( ps \/ ph ) )
4		dfordc 824	. . . . 5 |- (DECID ps -> ( ( ps \/ ph ) <-> ( -. ps -> ph ) ) )
5	3, 4	syl5rbb 191	. . . 4 |- (DECID ps -> ( ( -. ps -> ph ) <-> ( ph \/ ps ) ) )
6		imnan 656	. . . . 5 |- ( ( ph -> -. ps ) <-> -. ( ph /\ ps ) )
7	6	a1i 9	. . . 4 |- (DECID ps -> ( ( ph -> -. ps ) <-> -. ( ph /\ ps ) ) )
8	5, 7	anbi12d 456	. . 3 |- (DECID ps -> ( ( ( -. ps -> ph ) /\ ( ph -> -. ps ) ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) ) )
9	2, 8	syl5bb 190	. 2 |- (DECID ps -> ( ( -. ps <-> ph ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) ) )
10	1, 9	syl5rbb 191	1 |- (DECID ps -> ( ( ( ph \/ 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:  xor2dc  1321