Theorem indifdi 13823

Description: Distribute intersection over difference.

Assertion

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

Proof of Theorem indifdi

Step	Hyp	Ref	Expression
1		pm3.24 720	. . . . . . . 8 |- -. (x e. C /\ -. x e. C)
2	1	intnan 755	. . . . . . 7 |- -. (x e. A /\ (x e. C /\ -. x e. C))
3		anass 487	. . . . . . 7 |- (((x e. A /\ x e. C) /\ -. x e. C) <-> (x e. A /\ (x e. C /\ -. x e. C)))
4	2, 3	mtbir 209	. . . . . 6 |- -. ((x e. A /\ x e. C) /\ -. x e. C)
5	4	biorfi 808	. . . . 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		an23 543	. . . . 5 |- (((x e. A /\ -. x e. B) /\ x e. C) <-> ((x e. A /\ x e. C) /\ -. x e. B))
7		andi 665	. . . . 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 200	. . . 4 |- (((x e. A /\ -. x e. B) /\ x e. C) <-> ((x e. A /\ x e. C) /\ (-. x e. B \/ -. x e. C)))
9		ianor 329	. . . . 5 |- (-. (x e. B /\ x e. C) <-> (-. x e. B \/ -. x e. C))
10	9	anbi2i 538	. . . 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 193	. . 3 |- (((x e. A /\ -. x e. B) /\ x e. C) <-> ((x e. A /\ x e. C) /\ -. (x e. B /\ x e. C)))
12		elin 2786	. . . 4 |- (x e. ((A \ B) i^i C) <-> (x e. (A \ B) /\ x e. C))
13		eldif 2609	. . . . 5 |- (x e. (A \ B) <-> (x e. A /\ -. x e. B))
14	13	anbi1i 539	. . . 4 |- ((x e. (A \ B) /\ x e. C) <-> ((x e. A /\ -. x e. B) /\ x e. C))
15	12, 14	bitri 190	. . 3 |- (x e. ((A \ B) i^i C) <-> ((x e. A /\ -. x e. B) /\ x e. C))
16		eldif 2609	. . . 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 2786	. . . . 5 |- (x e. (A i^i C) <-> (x e. A /\ x e. C))
18		elin 2786	. . . . . 6 |- (x e. (B i^i C) <-> (x e. B /\ x e. C))
19	18	notbii 204	. . . . 5 |- (-. x e. (B i^i C) <-> -. (x e. B /\ x e. C))
20	17, 19	anbi12i 540	. . . 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 190	. . 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 200	. 2 |- (x e. ((A \ B) i^i C) <-> x e. ((A i^i C) \ (B i^i C)))
23	22	eqriv 1881	1 |- ((A \ B) i^i C) = ((A i^i C) \ (B i^i C))

Syntax hints:  -. wn 2   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300   \ cdif 2590   i^i cin 2592

This theorem is referenced by:  preddif 13902

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-dif 2597  df-in 2603

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