Theorem xordc 1283

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 1269	. . . 4 |- ( ( ph \/_ ps ) <-> ( ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) )
2		ancom 253	. . . . 5 |- ( ( -. ph /\ ps ) <-> ( ps /\ -. ph ) )
3	2	orbi2i 679	. . . 4 |- ( ( ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) <-> ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) )
4	1, 3	bitri 173	. . 3 |- ( ( ph \/_ ps ) <-> ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) )
5		xornbidc 1282	. . . 4 |- (DECID ph -> (DECID ps -> ( ( ph \/_ ps ) <-> -. ( ph <-> ps ) ) ) )
6	5	imp 115	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( ph \/_ ps ) <-> -. ( ph <-> ps ) ) )
7	4, 6	syl5rbbr 184	. 2 |- ( (DECID ph /\ DECID ps ) -> ( -. ( ph <-> ps ) <-> ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) ) )
8	7	ex 108	1 |- (DECID ph -> (DECID ps -> ( -. ( ph <-> ps ) <-> ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629  DECID wdc 742   \/_ wxo 1266

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743  df-xor 1267

This theorem is referenced by:  dfbi3dc  1288  pm5.24dc  1289