Theorem xordc 1323

Description: Two ways to express "exclusive or" between decidable propositions. Theorem *5.22 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.)

Assertion

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

Proof of Theorem xordc

Step	Hyp	Ref	Expression
1		excxor 1309	. . . 4 |- ( ( ph \/_ ps ) <-> ( ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) )
2		ancom 262	. . . . 5 |- ( ( -. ph /\ ps ) <-> ( ps /\ -. ph ) )
3	2	orbi2i 711	. . . 4 |- ( ( ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) <-> ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) )
4	1, 3	bitri 182	. . 3 |- ( ( ph \/_ ps ) <-> ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) )
5		xornbidc 1322	. . . 4 |- (DECID ph -> (DECID ps -> ( ( ph \/_ ps ) <-> -. ( ph <-> ps ) ) ) )
6	5	imp 122	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( ph \/_ ps ) <-> -. ( ph <-> ps ) ) )
7	4, 6	syl5rbbr 193	. 2 |- ( (DECID ph /\ DECID ps ) -> ( -. ( ph <-> ps ) <-> ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) ) )
8	7	ex 113	1 |- (DECID ph -> (DECID ps -> ( -. ( ph <-> ps ) <-> ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 661  DECID wdc 775   \/_ wxo 1306

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  df-xor 1307

This theorem is referenced by:  dfbi3dc  1328  pm5.24dc  1329