Theorem cbvdisjdavw2

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

	Ref	Expression
Hypotheses	cbvdisjdavw2.1	$$⊢\left(\phi \wedge x=y\right)\to C=D$$
	cbvdisjdavw2.2	$$⊢\left(\phi \wedge x=y\right)\to A=B$$
Assertion	cbvdisjdavw2	$$⊢\phi \to \left(\underset{x\in A}{Disj}C\leftrightarrow \underset{y\in B}{Disj}D\right)$$

Proof

Step	Hyp	Ref	Expression
1		cbvdisjdavw2.1	$$⊢\left(\phi \wedge x=y\right)\to C=D$$
2		cbvdisjdavw2.2	$$⊢\left(\phi \wedge x=y\right)\to A=B$$
3	1	eleq2d	$$⊢\left(\phi \wedge x=y\right)\to \left(t\in C\leftrightarrow t\in D\right)$$
4	3 2	cbvrmodavw2	$$⊢\phi \to \left({\exists }^{*}x\in At\in C\leftrightarrow {\exists }^{*}y\in Bt\in D\right)$$
5	4	albidv	$$⊢\phi \to \left(\forall t{\exists }^{*}x\in At\in C\leftrightarrow \forall t{\exists }^{*}y\in Bt\in D\right)$$
6		df-disj	$$⊢\underset{x\in A}{Disj}C\leftrightarrow \forall t{\exists }^{*}x\in At\in C$$
7		df-disj	$$⊢\underset{y\in B}{Disj}D\leftrightarrow \forall t{\exists }^{*}y\in Bt\in D$$
8	5 6 7	3bitr4g	$$⊢\phi \to \left(\underset{x\in A}{Disj}C\leftrightarrow \underset{y\in B}{Disj}D\right)$$