Theorem cbvrabdavw2

Description: Change bound variable and domain in restricted class abstractions. Deduction form. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypotheses	cbvrabdavw2.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
	cbvrabdavw2.2	$$⊢\left(\phi \wedge x=y\right)\to A=B$$
Assertion	cbvrabdavw2	$$⊢\phi \to \left\{x\in A|\psi \right\}=\left\{y\in B|\chi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		cbvrabdavw2.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
2		cbvrabdavw2.2	$$⊢\left(\phi \wedge x=y\right)\to A=B$$
3		eleq1w	$$⊢x=y\to \left(x\in A\leftrightarrow y\in A\right)$$
4	3	adantl	$$⊢\left(\phi \wedge x=y\right)\to \left(x\in A\leftrightarrow y\in A\right)$$
5	2	eleq2d	$$⊢\left(\phi \wedge x=y\right)\to \left(y\in A\leftrightarrow y\in B\right)$$
6	4 5	bitrd	$$⊢\left(\phi \wedge x=y\right)\to \left(x\in A\leftrightarrow y\in B\right)$$
7	6 1	anbi12d	$$⊢\left(\phi \wedge x=y\right)\to \left(\left(x\in A\wedge \psi \right)\leftrightarrow \left(y\in B\wedge \chi \right)\right)$$
8	7	cbvabdavw	$$⊢\phi \to \left\{x|\left(x\in A\wedge \psi \right)\right\}=\left\{y|\left(y\in B\wedge \chi \right)\right\}$$
9		df-rab	$$⊢\left\{x\in A|\psi \right\}=\left\{x|\left(x\in A\wedge \psi \right)\right\}$$
10		df-rab	$$⊢\left\{y\in B|\chi \right\}=\left\{y|\left(y\in B\wedge \chi \right)\right\}$$
11	8 9 10	3eqtr4g	$$⊢\phi \to \left\{x\in A|\psi \right\}=\left\{y\in B|\chi \right\}$$