Theorem xordc 1382

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		xornbidc 1381	. . . 4 |- (DECID ph -> (DECID ps -> ( ( ph \/_ ps ) <-> -. ( ph <-> ps ) ) ) )
2	1	imp 123	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( ph \/_ ps ) <-> -. ( ph <-> ps ) ) )
3		excxor 1368	. . . 4 |- ( ( ph \/_ ps ) <-> ( ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) )
4		ancom 264	. . . . 5 |- ( ( -. ph /\ ps ) <-> ( ps /\ -. ph ) )
5	4	orbi2i 752	. . . 4 |- ( ( ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) <-> ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) )
6	3, 5	bitri 183	. . 3 |- ( ( ph \/_ ps ) <-> ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) )
7	2, 6	bitr3di 194	. 2 |- ( (DECID ph /\ DECID ps ) -> ( -. ( ph <-> ps ) <-> ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) ) )
8	7	ex 114	1 |- (DECID ph -> (DECID ps -> ( -. ( ph <-> ps ) <-> ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 824   \/_ wxo 1365

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-stab 821  df-dc 825  df-xor 1366

This theorem is referenced by:  dfbi3dc  1387  pm5.24dc  1388