Theorem cbvdisjv

Description: Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016)

	Ref	Expression
Hypothesis	cbvdisjv.1	|- ( x = y -> B = C )
Assertion	cbvdisjv	|- ( Disj_ x e. A B <-> Disj_ y e. A C )

Proof

Step	Hyp	Ref	Expression
1		cbvdisjv.1	|- ( x = y -> B = C )
2	1	eleq2d	|- ( x = y -> ( z e. B <-> z e. C ) )
3	2	cbvrmovw	|- ( E* x e. A z e. B <-> E* y e. A z e. C )
4	3	albii	|- ( A. z E* x e. A z e. B <-> A. z E* y e. A z e. C )
5		df-disj	|- ( Disj_ x e. A B <-> A. z E* x e. A z e. B )
6		df-disj	|- ( Disj_ y e. A C <-> A. z E* y e. A z e. C )
7	4 5 6	3bitr4i	|- ( Disj_ x e. A B <-> Disj_ y e. A C )