Theorem cbviunf 29372

Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.)

Hypotheses

Ref	Expression
cbviunf.x	|- F/_ x A
cbviunf.y	|- F/_ y A
cbviunf.1	|- F/_ y B
cbviunf.2	|- F/_ x C
cbviunf.3	|- ( x = y -> B = C )

Assertion

Ref	Expression
cbviunf	|- U_ x e. A B = U_ y e. A C

Distinct variable group:   x, y

Allowed substitution hints:   A( x, y)   B( x, y)   C( x, y)

Proof of Theorem cbviunf

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbviunf.x	. . . 4 |- F/_ x A
2		cbviunf.y	. . . 4 |- F/_ y A
3		cbviunf.1	. . . . 5 |- F/_ y B
4	3	nfcri 2758	. . . 4 |- F/ y z e. B
5		cbviunf.2	. . . . 5 |- F/_ x C
6	5	nfcri 2758	. . . 4 |- F/ x z e. C
7		cbviunf.3	. . . . 5 |- ( x = y -> B = C )
8	7	eleq2d 2687	. . . 4 |- ( x = y -> ( z e. B <-> z e. C ) )
9	1, 2, 4, 6, 8	cbvrexf 3166	. . 3 |- ( E. x e. A z e. B <-> E. y e. A z e. C )
10	9	abbii 2739	. 2 |- { z | E. x e. A z e. B } = { z | E. y e. A z e. C }
11		df-iun 4522	. 2 |- U_ x e. A B = { z | E. x e. A z e. B }
12		df-iun 4522	. 2 |- U_ y e. A C = { z | E. y e. A z e. C }
13	10, 11, 12	3eqtr4i 2654	1 |- U_ x e. A B = U_ y e. A C

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {cab 2608   F/_wnfc 2751   E.wrex 2913   U_ciun 4520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-iun 4522

This theorem is referenced by:  aciunf1lem  29462