Theorem xornbidc 1391

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		df-xor 1376	. . 3 |- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) )
2		xor2dc 1390	. . . 4 |- (DECID ph -> (DECID ps -> ( -. ( ph <-> ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) ) ) )
3	2	imp 124	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( -. ( ph <-> ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) ) )
4	1, 3	bitr4id 199	. 2 |- ( (DECID ph /\ DECID ps ) -> ( ( ph \/_ ps ) <-> -. ( ph <-> ps ) ) )
5	4	ex 115	1 |- (DECID ph -> (DECID ps -> ( ( ph \/_ ps ) <-> -. ( ph <-> ps ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708  DECID wdc 834   \/_ wxo 1375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-xor 1376

This theorem is referenced by:  xordc  1392  xordidc  1399