Theorem cbviinv

Description: Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009) Add disjoint variable condition to avoid ax-13 . See cbviinvg for a less restrictive version requiring more axioms. (Revised by GG, 14-Aug-2025)

	Ref	Expression
Hypothesis	cbviunv.1	|- ( x = y -> B = C )
Assertion	cbviinv	|- |^|_ x e. A B = |^|_ y e. A C

Proof

Step	Hyp	Ref	Expression
1		cbviunv.1	|- ( x = y -> B = C )
2	1	eleq2d	|- ( x = y -> ( z e. B <-> z e. C ) )
3	2	cbvralvw	|- ( A. x e. A z e. B <-> A. y e. A z e. C )
4	3	abbii	|- { z | A. x e. A z e. B } = { z | A. y e. A z e. C }
5		df-iin	|- |^|_ x e. A B = { z | A. x e. A z e. B }
6		df-iin	|- |^|_ y e. A C = { z | A. y e. A z e. C }
7	4 5 6	3eqtr4i	|- |^|_ x e. A B = |^|_ y e. A C