Theorem cbviotadavw

Description: Change bound variable in a description binder. Deduction form. (Contributed by GG, 14-Aug-2025)

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

Proof

Step	Hyp	Ref	Expression
1		cbviotadavw.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	eqeq1d	$$⊢\phi \to \left(\left\{x|\psi \right\}=\left\{t\right\}\leftrightarrow \left\{y|\chi \right\}=\left\{t\right\}\right)$$
4	3	abbidv	$$⊢\phi \to \left\{t|\left\{x|\psi \right\}=\left\{t\right\}\right\}=\left\{t|\left\{y|\chi \right\}=\left\{t\right\}\right\}$$
5	4	unieqd	$$⊢\phi \to \bigcup \left\{t|\left\{x|\psi \right\}=\left\{t\right\}\right\}=\bigcup \left\{t|\left\{y|\chi \right\}=\left\{t\right\}\right\}$$
6		df-iota	$$⊢\left(\iota x|\psi \right)=\bigcup \left\{t|\left\{x|\psi \right\}=\left\{t\right\}\right\}$$
7		df-iota	$$⊢\left(\iota y|\chi \right)=\bigcup \left\{t|\left\{y|\chi \right\}=\left\{t\right\}\right\}$$
8	5 6 7	3eqtr4g	$$⊢\phi \to \left(\iota x|\psi \right)=\left(\iota y|\chi \right)$$