Theorem cbvdisjv

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

	Ref	Expression
Hypothesis	cbvdisjv.1	$$⊢x=y\to B=C$$
Assertion	cbvdisjv	$$⊢\underset{x\in A}{Disj}B\leftrightarrow \underset{y\in A}{Disj}C$$

Proof

Step	Hyp	Ref	Expression
1		cbvdisjv.1	$$⊢x=y\to B=C$$
2	1	eleq2d	$$⊢x=y\to \left(z\in B\leftrightarrow z\in C\right)$$
3	2	cbvrmovw	$$⊢{\exists }^{*}x\in Az\in B\leftrightarrow {\exists }^{*}y\in Az\in C$$
4	3	albii	$$⊢\forall z{\exists }^{*}x\in Az\in B\leftrightarrow \forall z{\exists }^{*}y\in Az\in C$$
5		df-disj	$$⊢\underset{x\in A}{Disj}B\leftrightarrow \forall z{\exists }^{*}x\in Az\in B$$
6		df-disj	$$⊢\underset{y\in A}{Disj}C\leftrightarrow \forall z{\exists }^{*}y\in Az\in C$$
7	4 5 6	3bitr4i	$$⊢\underset{x\in A}{Disj}B\leftrightarrow \underset{y\in A}{Disj}C$$