Theorem cbvabdavw

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

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

Proof

Step	Hyp	Ref	Expression
1		cbvabdavw.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
2	1	cbvsbdavw	$$⊢\phi \to \left(\left[t/x\right]\psi \leftrightarrow \left[t/y\right]\chi \right)$$
3		df-clab	$$⊢t\in \left\{x|\psi \right\}\leftrightarrow \left[t/x\right]\psi$$
4		df-clab	$$⊢t\in \left\{y|\chi \right\}\leftrightarrow \left[t/y\right]\chi$$
5	2 3 4	3bitr4g	$$⊢\phi \to \left(t\in \left\{x|\psi \right\}\leftrightarrow t\in \left\{y|\chi \right\}\right)$$
6	5	eqrdv	$$⊢\phi \to \left\{x|\psi \right\}=\left\{y|\chi \right\}$$