Theorem cbviin

Description: Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009) (Revised by Mario Carneiro, 14-Oct-2016) Add disjoint variable condition to avoid ax-13 . See cbviing for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024)

	Ref	Expression
Hypotheses	cbviun.1	|- F/_ y B
	cbviun.2	|- F/_ x C
	cbviun.3	|- ( x = y -> B = C )
Assertion	cbviin	|- |^|_ x e. A B = |^|_ y e. A C

Proof

Step	Hyp	Ref	Expression
1		cbviun.1	|- F/_ y B
2		cbviun.2	|- F/_ x C
3		cbviun.3	|- ( x = y -> B = C )
4	1	nfcri	|- F/ y z e. B
5	2	nfcri	|- F/ x z e. C
6	3	eleq2d	|- ( x = y -> ( z e. B <-> z e. C ) )
7	4 5 6	cbvralw	|- ( A. x e. A z e. B <-> A. y e. A z e. C )
8	7	abbii	|- { z | A. x e. A z e. B } = { z | A. y e. A z e. C }
9		df-iin	|- |^|_ x e. A B = { z | A. x e. A z e. B }
10		df-iin	|- |^|_ y e. A C = { z | A. y e. A z e. C }
11	8 9 10	3eqtr4i	|- |^|_ x e. A B = |^|_ y e. A C