Theorem xornbidc 1322

Description: Exclusive or is equivalent to negated biconditional for decidable propositions. (Contributed by Jim Kingdon, 27-Apr-2018.)

Assertion

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

Proof of Theorem xornbidc

Step	Hyp	Ref	Expression
1		xor2dc 1321	. . . 4 |- (DECID ph -> (DECID ps -> ( -. ( ph <-> ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) ) ) )
2	1	imp 122	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( -. ( ph <-> ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) ) )
3		df-xor 1307	. . 3 |- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) )
4	2, 3	syl6rbbr 197	. 2 |- ( (DECID ph /\ DECID ps ) -> ( ( ph \/_ ps ) <-> -. ( ph <-> ps ) ) )
5	4	ex 113	1 |- (DECID ph -> (DECID ps -> ( ( ph \/_ ps ) <-> -. ( ph <-> ps ) ) ) )

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:  xordc  1323  xordidc  1330