Theorem annimdc 939

Description: Express conjunction in terms of implication. The forward direction, annimim 687, is valid for all propositions, but as an equivalence, it requires a decidability condition. (Contributed by Jim Kingdon, 25-Apr-2018.)

Assertion

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

Proof of Theorem annimdc

Step	Hyp	Ref	Expression
1		imandc 890	. . . 4 |- (DECID ps -> ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) )
2	1	adantl 277	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) )
3		dcim 842	. . . . 5 |- (DECID ph -> (DECID ps -> DECID ( ph -> ps ) ) )
4	3	imp 124	. . . 4 |- ( (DECID ph /\ DECID ps ) -> DECID ( ph -> ps ) )
5		dcn 843	. . . . 5 |- (DECID ps -> DECID -. ps )
6		dcan 935	. . . . 5 |- ( (DECID ph /\ DECID -. ps ) -> DECID ( ph /\ -. ps ) )
7	5, 6	sylan2 286	. . . 4 |- ( (DECID ph /\ DECID ps ) -> DECID ( ph /\ -. ps ) )
8		con2bidc 876	. . . 4 |- (DECID ( ph -> ps ) -> (DECID ( ph /\ -. ps ) -> ( ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) <-> ( ( ph /\ -. ps ) <-> -. ( ph -> ps ) ) ) ) )
9	4, 7, 8	sylc 62	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) <-> ( ( ph /\ -. ps ) <-> -. ( ph -> ps ) ) ) )
10	2, 9	mpbid 147	. 2 |- ( (DECID ph /\ DECID ps ) -> ( ( ph /\ -. ps ) <-> -. ( ph -> ps ) ) )
11	10	ex 115	1 |- (DECID ph -> (DECID ps -> ( ( ph /\ -. ps ) <-> -. ( ph -> ps ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 835

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 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836

This theorem is referenced by:  xordidc  1410