Theorem xordidc 1389

Description: Conjunction distributes over exclusive-or, for decidable propositions. This is one way to interpret the distributive law of multiplication over addition in modulo 2 arithmetic. (Contributed by Jim Kingdon, 14-Jul-2018.)

Assertion

Ref	Expression
xordidc	|- (DECID ph -> (DECID ps -> (DECID ch -> ( ( ph /\ ( ps \/_ ch ) ) <-> ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) ) ) ) )

Proof of Theorem xordidc

Step	Hyp	Ref	Expression
1		dcbi 926	. . . . 5 |- (DECID ps -> (DECID ch -> DECID ( ps <-> ch ) ) )
2	1	imp 123	. . . 4 |- ( (DECID ps /\ DECID ch ) -> DECID ( ps <-> ch ) )
3		annimdc 927	. . . . . 6 |- (DECID ph -> (DECID ( ps <-> ch ) -> ( ( ph /\ -. ( ps <-> ch ) ) <-> -. ( ph -> ( ps <-> ch ) ) ) ) )
4	3	imp 123	. . . . 5 |- ( (DECID ph /\ DECID ( ps <-> ch ) ) -> ( ( ph /\ -. ( ps <-> ch ) ) <-> -. ( ph -> ( ps <-> ch ) ) ) )
5		pm5.32 449	. . . . . 6 |- ( ( ph -> ( ps <-> ch ) ) <-> ( ( ph /\ ps ) <-> ( ph /\ ch ) ) )
6	5	notbii 658	. . . . 5 |- ( -. ( ph -> ( ps <-> ch ) ) <-> -. ( ( ph /\ ps ) <-> ( ph /\ ch ) ) )
7	4, 6	bitrdi 195	. . . 4 |- ( (DECID ph /\ DECID ( ps <-> ch ) ) -> ( ( ph /\ -. ( ps <-> ch ) ) <-> -. ( ( ph /\ ps ) <-> ( ph /\ ch ) ) ) )
8	2, 7	sylan2 284	. . 3 |- ( (DECID ph /\ (DECID ps /\ DECID ch ) ) -> ( ( ph /\ -. ( ps <-> ch ) ) <-> -. ( ( ph /\ ps ) <-> ( ph /\ ch ) ) ) )
9		xornbidc 1381	. . . . . 6 |- (DECID ps -> (DECID ch -> ( ( ps \/_ ch ) <-> -. ( ps <-> ch ) ) ) )
10	9	imp 123	. . . . 5 |- ( (DECID ps /\ DECID ch ) -> ( ( ps \/_ ch ) <-> -. ( ps <-> ch ) ) )
11	10	adantl 275	. . . 4 |- ( (DECID ph /\ (DECID ps /\ DECID ch ) ) -> ( ( ps \/_ ch ) <-> -. ( ps <-> ch ) ) )
12	11	anbi2d 460	. . 3 |- ( (DECID ph /\ (DECID ps /\ DECID ch ) ) -> ( ( ph /\ ( ps \/_ ch ) ) <-> ( ph /\ -. ( ps <-> ch ) ) ) )
13		dcan2 924	. . . . . 6 |- (DECID ph -> (DECID ps -> DECID ( ph /\ ps ) ) )
14	13	imp 123	. . . . 5 |- ( (DECID ph /\ DECID ps ) -> DECID ( ph /\ ps ) )
15	14	adantrr 471	. . . 4 |- ( (DECID ph /\ (DECID ps /\ DECID ch ) ) -> DECID ( ph /\ ps ) )
16		dcan2 924	. . . . . 6 |- (DECID ph -> (DECID ch -> DECID ( ph /\ ch ) ) )
17	16	imp 123	. . . . 5 |- ( (DECID ph /\ DECID ch ) -> DECID ( ph /\ ch ) )
18	17	adantrl 470	. . . 4 |- ( (DECID ph /\ (DECID ps /\ DECID ch ) ) -> DECID ( ph /\ ch ) )
19		xornbidc 1381	. . . 4 |- (DECID ( ph /\ ps ) -> (DECID ( ph /\ ch ) -> ( ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) <-> -. ( ( ph /\ ps ) <-> ( ph /\ ch ) ) ) ) )
20	15, 18, 19	sylc 62	. . 3 |- ( (DECID ph /\ (DECID ps /\ DECID ch ) ) -> ( ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) <-> -. ( ( ph /\ ps ) <-> ( ph /\ ch ) ) ) )
21	8, 12, 20	3bitr4d 219	. 2 |- ( (DECID ph /\ (DECID ps /\ DECID ch ) ) -> ( ( ph /\ ( ps \/_ ch ) ) <-> ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) ) )
22	21	exp32 363	1 |- (DECID ph -> (DECID ps -> (DECID ch -> ( ( ph /\ ( ps \/_ ch ) ) <-> ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104  DECID wdc 824   \/_ wxo 1365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-xor 1366

This theorem is referenced by: (None)