Theorem xordidc 1290

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 844	. . . . 5 |- (DECID ps -> (DECID ch -> DECID ( ps <-> ch ) ) )
2	1	imp 115	. . . 4 |- ( (DECID ps /\ DECID ch ) -> DECID ( ps <-> ch ) )
3		annimdc 845	. . . . . 6 |- (DECID ph -> (DECID ( ps <-> ch ) -> ( ( ph /\ -. ( ps <-> ch ) ) <-> -. ( ph -> ( ps <-> ch ) ) ) ) )
4	3	imp 115	. . . . 5 |- ( (DECID ph /\ DECID ( ps <-> ch ) ) -> ( ( ph /\ -. ( ps <-> ch ) ) <-> -. ( ph -> ( ps <-> ch ) ) ) )
5		pm5.32 426	. . . . . 6 |- ( ( ph -> ( ps <-> ch ) ) <-> ( ( ph /\ ps ) <-> ( ph /\ ch ) ) )
6	5	notbii 594	. . . . 5 |- ( -. ( ph -> ( ps <-> ch ) ) <-> -. ( ( ph /\ ps ) <-> ( ph /\ ch ) ) )
7	4, 6	syl6bb 185	. . . 4 |- ( (DECID ph /\ DECID ( ps <-> ch ) ) -> ( ( ph /\ -. ( ps <-> ch ) ) <-> -. ( ( ph /\ ps ) <-> ( ph /\ ch ) ) ) )
8	2, 7	sylan2 270	. . 3 |- ( (DECID ph /\ (DECID ps /\ DECID ch ) ) -> ( ( ph /\ -. ( ps <-> ch ) ) <-> -. ( ( ph /\ ps ) <-> ( ph /\ ch ) ) ) )
9		xornbidc 1282	. . . . . 6 |- (DECID ps -> (DECID ch -> ( ( ps \/_ ch ) <-> -. ( ps <-> ch ) ) ) )
10	9	imp 115	. . . . 5 |- ( (DECID ps /\ DECID ch ) -> ( ( ps \/_ ch ) <-> -. ( ps <-> ch ) ) )
11	10	adantl 262	. . . 4 |- ( (DECID ph /\ (DECID ps /\ DECID ch ) ) -> ( ( ps \/_ ch ) <-> -. ( ps <-> ch ) ) )
12	11	anbi2d 437	. . 3 |- ( (DECID ph /\ (DECID ps /\ DECID ch ) ) -> ( ( ph /\ ( ps \/_ ch ) ) <-> ( ph /\ -. ( ps <-> ch ) ) ) )
13		dcan 842	. . . . . 6 |- (DECID ph -> (DECID ps -> DECID ( ph /\ ps ) ) )
14	13	imp 115	. . . . 5 |- ( (DECID ph /\ DECID ps ) -> DECID ( ph /\ ps ) )
15	14	adantrr 448	. . . 4 |- ( (DECID ph /\ (DECID ps /\ DECID ch ) ) -> DECID ( ph /\ ps ) )
16		dcan 842	. . . . . 6 |- (DECID ph -> (DECID ch -> DECID ( ph /\ ch ) ) )
17	16	imp 115	. . . . 5 |- ( (DECID ph /\ DECID ch ) -> DECID ( ph /\ ch ) )
18	17	adantrl 447	. . . 4 |- ( (DECID ph /\ (DECID ps /\ DECID ch ) ) -> DECID ( ph /\ ch ) )
19		xornbidc 1282	. . . 4 |- (DECID ( ph /\ ps ) -> (DECID ( ph /\ ch ) -> ( ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) <-> -. ( ( ph /\ ps ) <-> ( ph /\ ch ) ) ) ) )
20	15, 18, 19	sylc 56	. . 3 |- ( (DECID ph /\ (DECID ps /\ DECID ch ) ) -> ( ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) <-> -. ( ( ph /\ ps ) <-> ( ph /\ ch ) ) ) )
21	8, 12, 20	3bitr4d 209	. 2 |- ( (DECID ph /\ (DECID ps /\ DECID ch ) ) -> ( ( ph /\ ( ps \/_ ch ) ) <-> ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) ) )
22	21	exp32 347	1 |- (DECID ph -> (DECID ps -> (DECID ch -> ( ( ph /\ ( ps \/_ ch ) ) <-> ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98  DECID wdc 742   \/_ wxo 1266

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  df-xor 1267

This theorem is referenced by: (None)