Theorem cbvsbcdavw

Description: Change bound variable of a class substitution. Deduction form. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypothesis	cbvsbcdavw.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	cbvsbcdavw	$$⊢\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \leftrightarrow \underset{\dot{}}{[}A/y\underset{\dot{}}{]}\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		cbvsbcdavw.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
2	1	cbvabdavw	$$⊢\phi \to \left\{x|\psi \right\}=\left\{y|\chi \right\}$$
3	2	eleq2d	$$⊢\phi \to \left(A\in \left\{x|\psi \right\}\leftrightarrow A\in \left\{y|\chi \right\}\right)$$
4		df-sbc	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \leftrightarrow A\in \left\{x|\psi \right\}$$
5		df-sbc	$$⊢\underset{\dot{}}{[}A/y\underset{\dot{}}{]}\chi \leftrightarrow A\in \left\{y|\chi \right\}$$
6	3 4 5	3bitr4g	$$⊢\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \leftrightarrow \underset{\dot{}}{[}A/y\underset{\dot{}}{]}\chi \right)$$