Theorem cbviunf

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)

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

Proof

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