Theorem intun 4319

Description: The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42. (Contributed by NM, 22-Sep-2002.)

Assertion

Ref	Expression
intun

Proof of Theorem intun

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.26 1680	. . . 4
2		elun 3644	. . . . . . 7
3	2	imbi1i 325	. . . . . 6
4		jaob 783	. . . . . 6
5	3, 4	bitri 249	. . . . 5
6	5	albii 1640	. . . 4
7		vex 3112	. . . . . 6
8	7	elint 4292	. . . . 5
9	7	elint 4292	. . . . 5
10	8, 9	anbi12i 697	. . . 4
11	1, 6, 10	3bitr4i 277	. . 3
12	7	elint 4292	. . 3
13		elin 3686	. . 3
14	11, 12, 13	3bitr4i 277	. 2
15	14	eqriv 2453	1

Syntax hints:  ->wi 4  \/wo 368  /\wa 369  A.wal 1393  =wceq 1395  e.wcel 1818  u.cun 3473  i^icin 3474  |^|cint 4286

This theorem is referenced by:  intunsn  4326  riinint  5264  fiin  7902  elfiun  7910  elrfi  30626

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-un 3480  df-in 3482  df-int 4287