Theorem pm5.24dc 1329

Description: Theorem *5.24 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.)

Assertion

Ref	Expression
pm5.24dc	|- (DECID ph -> (DECID ps -> ( -. ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) ) <-> ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) ) ) )

Proof of Theorem pm5.24dc

Step	Hyp	Ref	Expression
1		dfbi3dc 1328	. . . . 5 |- (DECID ph -> (DECID ps -> ( ( ph <-> ps ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) ) ) ) )
2	1	imp 122	. . . 4 |- ( (DECID ph /\ DECID ps ) -> ( ( ph <-> ps ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) ) ) )
3	2	notbid 624	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( -. ( ph <-> ps ) <-> -. ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) ) ) )
4		xordc 1323	. . . 4 |- (DECID ph -> (DECID ps -> ( -. ( ph <-> ps ) <-> ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) ) ) )
5	4	imp 122	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( -. ( ph <-> ps ) <-> ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) ) )
6	3, 5	bitr3d 188	. 2 |- ( (DECID ph /\ DECID ps ) -> ( -. ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) ) <-> ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) ) )
7	6	ex 113	1 |- (DECID ph -> (DECID ps -> ( -. ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) ) <-> ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) ) ) )

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

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: (None)