Theorem iuneq12df 4354

Description: Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 31-Dec-2016.)

Hypotheses

Ref	Expression
iuneq12df.1
iuneq12df.2
iuneq12df.3
iuneq12df.4
iuneq12df.5

Assertion

Ref	Expression
iuneq12df

Proof of Theorem iuneq12df

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iuneq12df.1	. . . 4
2		iuneq12df.2	. . . 4
3		iuneq12df.3	. . . 4
4		iuneq12df.4	. . . 4
5		iuneq12df.5	. . . . 5
6	5	eleq2d 2527	. . . 4
7	1, 2, 3, 4, 6	rexeqbid 3067	. . 3
8	7	alrimiv 1719	. 2
9		abbi 2588	. . 3
10		df-iun 4332	. . . 4
11		df-iun 4332	. . . 4
12	10, 11	eqeq12i 2477	. . 3
13	9, 12	bitr4i 252	. 2
14	8, 13	sylib 196	1

Syntax hints:  ->wi 4  <->wb 184  A.wal 1393  =wceq 1395  F/wnf 1616  e.wcel 1818  {cab 2442  F/_wnfc 2605  E.wrex 2808  U_ciun 4330

This theorem is referenced by:  iundisjf  27448  measvuni  28185  iuneq2f  30563

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-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-rex 2813  df-iun 4332