Theorem iunxiun 4413

Description: Separate an indexed union in the index of an indexed union. (Contributed by Mario Carneiro, 5-Dec-2016.)

Assertion

Ref	Expression
iunxiun

Distinct variable groups:   ,   ,   ,

Proof of Theorem iunxiun

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 4335	. . . . . . . 8
2	1	anbi1i 695	. . . . . . 7
3		r19.41v 3009	. . . . . . 7
4	2, 3	bitr4i 252	. . . . . 6
5	4	exbii 1667	. . . . 5
6		rexcom4 3129	. . . . 5
7	5, 6	bitr4i 252	. . . 4
8		df-rex 2813	. . . 4
9		eliun 4335	. . . . . 6
10		df-rex 2813	. . . . . 6
11	9, 10	bitri 249	. . . . 5
12	11	rexbii 2959	. . . 4
13	7, 8, 12	3bitr4i 277	. . 3
14		eliun 4335	. . 3
15		eliun 4335	. . 3
16	13, 14, 15	3bitr4i 277	. 2
17	16	eqriv 2453	1

Syntax hints:  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  E.wrex 2808  U_ciun 4330

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-ral 2812  df-rex 2813  df-v 3111  df-iun 4332