Theorem cbvdisjdavw

Description: Change bound variable in a disjoint collection. Deduction form. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypothesis	cbvdisjdavw.1	$$⊢\left(\phi \wedge x=y\right)\to B=C$$
Assertion	cbvdisjdavw	$$⊢\phi \to \left(\underset{x\in A}{Disj}B\leftrightarrow \underset{y\in A}{Disj}C\right)$$

Proof

Step	Hyp	Ref	Expression
1		cbvdisjdavw.1	$$⊢\left(\phi \wedge x=y\right)\to B=C$$
2	1	eleq2d	$$⊢\left(\phi \wedge x=y\right)\to \left(t\in B\leftrightarrow t\in C\right)$$
3	2	cbvrmodavw	$$⊢\phi \to \left({\exists }^{*}x\in At\in B\leftrightarrow {\exists }^{*}y\in At\in C\right)$$
4	3	albidv	$$⊢\phi \to \left(\forall t{\exists }^{*}x\in At\in B\leftrightarrow \forall t{\exists }^{*}y\in At\in C\right)$$
5		df-disj	$$⊢\underset{x\in A}{Disj}B\leftrightarrow \forall t{\exists }^{*}x\in At\in B$$
6		df-disj	$$⊢\underset{y\in A}{Disj}C\leftrightarrow \forall t{\exists }^{*}y\in At\in C$$
7	4 5 6	3bitr4g	$$⊢\phi \to \left(\underset{x\in A}{Disj}B\leftrightarrow \underset{y\in A}{Disj}C\right)$$