Theorem indifdir 3512

Description: Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)

Assertion

Ref	Expression
indifdir	|- ((A \ B) i^i C) = ((A i^i C) \ (B i^i C))

Proof of Theorem indifdir

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm3.24 852	. . . . . . . 8 |- -. (x e. C /\ -. x e. C)
2	1	intnan 880	. . . . . . 7 |- -. (x e. A /\ (x e. C /\ -. x e. C))
3		anass 630	. . . . . . 7 |- (((x e. A /\ x e. C) /\ -. x e. C) <-> (x e. A /\ (x e. C /\ -. x e. C)))
4	2, 3	mtbir 290	. . . . . 6 |- -. ((x e. A /\ x e. C) /\ -. x e. C)
5	4	biorfi 396	. . . . 5 |- (((x e. A /\ x e. C) /\ -. x e. B) <-> (((x e. A /\ x e. C) /\ -. x e. B) \/ ((x e. A /\ x e. C) /\ -. x e. C)))
6		an32 773	. . . . 5 |- (((x e. A /\ -. x e. B) /\ x e. C) <-> ((x e. A /\ x e. C) /\ -. x e. B))
7		andi 837	. . . . 5 |- (((x e. A /\ x e. C) /\ (-. x e. B \/ -. x e. C)) <-> (((x e. A /\ x e. C) /\ -. x e. B) \/ ((x e. A /\ x e. C) /\ -. x e. C)))
8	5, 6, 7	3bitr4i 268	. . . 4 |- (((x e. A /\ -. x e. B) /\ x e. C) <-> ((x e. A /\ x e. C) /\ (-. x e. B \/ -. x e. C)))
9		ianor 474	. . . . 5 |- (-. (x e. B /\ x e. C) <-> (-. x e. B \/ -. x e. C))
10	9	anbi2i 675	. . . 4 |- (((x e. A /\ x e. C) /\ -. (x e. B /\ x e. C)) <-> ((x e. A /\ x e. C) /\ (-. x e. B \/ -. x e. C)))
11	8, 10	bitr4i 243	. . 3 |- (((x e. A /\ -. x e. B) /\ x e. C) <-> ((x e. A /\ x e. C) /\ -. (x e. B /\ x e. C)))
12		elin 3220	. . . 4 |- (x e. ((A \ B) i^i C) <-> (x e. (A \ B) /\ x e. C))
13		eldif 3222	. . . . 5 |- (x e. (A \ B) <-> (x e. A /\ -. x e. B))
14	13	anbi1i 676	. . . 4 |- ((x e. (A \ B) /\ x e. C) <-> ((x e. A /\ -. x e. B) /\ x e. C))
15	12, 14	bitri 240	. . 3 |- (x e. ((A \ B) i^i C) <-> ((x e. A /\ -. x e. B) /\ x e. C))
16		eldif 3222	. . . 4 |- (x e. ((A i^i C) \ (B i^i C)) <-> (x e. (A i^i C) /\ -. x e. (B i^i C)))
17		elin 3220	. . . . 5 |- (x e. (A i^i C) <-> (x e. A /\ x e. C))
18		elin 3220	. . . . . 6 |- (x e. (B i^i C) <-> (x e. B /\ x e. C))
19	18	notbii 287	. . . . 5 |- (-. x e. (B i^i C) <-> -. (x e. B /\ x e. C))
20	17, 19	anbi12i 678	. . . 4 |- ((x e. (A i^i C) /\ -. x e. (B i^i C)) <-> ((x e. A /\ x e. C) /\ -. (x e. B /\ x e. C)))
21	16, 20	bitri 240	. . 3 |- (x e. ((A i^i C) \ (B i^i C)) <-> ((x e. A /\ x e. C) /\ -. (x e. B /\ x e. C)))
22	11, 15, 21	3bitr4i 268	. 2 |- (x e. ((A \ B) i^i C) <-> x e. ((A i^i C) \ (B i^i C)))
23	22	eqriv 2350	1 |- ((A \ B) i^i C) = ((A i^i C) \ (B i^i C))

Syntax hints:  -. wn 3   \/ wo 357   /\ wa 358   = wceq 1642   e. wcel 1710   \ cdif 3207   i^i cin 3209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216

This theorem is referenced by: (None)