Theorem iuncom 4337

Description: Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)

Assertion

Ref	Expression
iuncom

Distinct variable groups:   ,   ,   ,

Proof of Theorem iuncom

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexcom 3019	. . . 4
2		eliun 4335	. . . . 5
3	2	rexbii 2959	. . . 4
4		eliun 4335	. . . . 5
5	4	rexbii 2959	. . . 4
6	1, 3, 5	3bitr4i 277	. . 3
7		eliun 4335	. . 3
8		eliun 4335	. . 3
9	6, 7, 8	3bitr4i 277	. 2
10	9	eqriv 2453	1

Syntax hints:  =wceq 1395  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