Theorem cbvrabdavw

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

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

Proof

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