Theorem xordidc 1287

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 843	. . . . 5 |- (DECID ps -> (DECID ch -> DECID (ps <-> ch)))
2	1	imp 115	. . . 4 |- ((DECID ps /\ DECID ch) -> DECID (ps <-> ch))
3		annimdc 844	. . . . . 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 593	. . . . 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 1279	. . . . . 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 841	. . . . . 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 841	. . . . . 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 1279	. . . 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 741   \/_ wxo 1265

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 629

This theorem depends on definitions:  df-bi 110  df-dc 742  df-xor 1266

This theorem is referenced by: (None)