Theorem annimdc 845

Description: Express conjunction in terms of implication. The forward direction, annimim 782, 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 786	. . . 4 |- (DECID ps -> ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) )
2	1	adantl 262	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) )
3		dcim 784	. . . . 5 |- (DECID ph -> (DECID ps -> DECID ( ph -> ps ) ) )
4	3	imp 115	. . . 4 |- ( (DECID ph /\ DECID ps ) -> DECID ( ph -> ps ) )
5		dcn 746	. . . . . 6 |- (DECID ps -> DECID -. ps )
6		dcan 842	. . . . . 6 |- (DECID ph -> (DECID -. ps -> DECID ( ph /\ -. ps ) ) )
7	5, 6	syl5 28	. . . . 5 |- (DECID ph -> (DECID ps -> DECID ( ph /\ -. ps ) ) )
8	7	imp 115	. . . 4 |- ( (DECID ph /\ DECID ps ) -> DECID ( ph /\ -. ps ) )
9		con2bidc 769	. . . 4 |- (DECID ( ph -> ps ) -> (DECID ( ph /\ -. ps ) -> ( ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) <-> ( ( ph /\ -. ps ) <-> -. ( ph -> ps ) ) ) ) )
10	4, 8, 9	sylc 56	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) <-> ( ( ph /\ -. ps ) <-> -. ( ph -> ps ) ) ) )
11	2, 10	mpbid 135	. 2 |- ( (DECID ph /\ DECID ps ) -> ( ( ph /\ -. ps ) <-> -. ( ph -> ps ) ) )
12	11	ex 108	1 |- (DECID ph -> (DECID ps -> ( ( ph /\ -. ps ) <-> -. ( ph -> ps ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by:  xordidc  1290