Theorem indifdir 3753

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

Assertion

Ref	Expression
indifdir

Proof of Theorem indifdir

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm3.24 882	. . . . . . . 8
2	1	intnan 914	. . . . . . 7
3		anass 649	. . . . . . 7
4	2, 3	mtbir 299	. . . . . 6
5	4	biorfi 407	. . . . 5
6		an32 798	. . . . 5
7		andi 867	. . . . 5
8	5, 6, 7	3bitr4i 277	. . . 4
9		ianor 488	. . . . 5
10	9	anbi2i 694	. . . 4
11	8, 10	bitr4i 252	. . 3
12		elin 3686	. . . 4
13		eldif 3485	. . . . 5
14	13	anbi1i 695	. . . 4
15	12, 14	bitri 249	. . 3
16		eldif 3485	. . . 4
17		elin 3686	. . . . 5
18		elin 3686	. . . . . 6
19	18	notbii 296	. . . . 5
20	17, 19	anbi12i 697	. . . 4
21	16, 20	bitri 249	. . 3
22	11, 15, 21	3bitr4i 277	. 2
23	22	eqriv 2453	1

Syntax hints:  -.wn 3  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  \cdif 3472  i^icin 3474

This theorem is referenced by:  fresaun  5761  uniioombllem4  21995  preddif  29271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-dif 3478  df-in 3482