Theorem cbviotavw

Description: Change bound variables in a description binder. Version of cbviotav with a disjoint variable condition, which requires fewer axioms . (Contributed by Andrew Salmon, 1-Aug-2011) (Revised by Gino Giotto, 30-Sep-2024)

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

Proof

Step	Hyp	Ref	Expression
1		cbviotavw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2	1	cbvabv	$$⊢\left\{x|\phi \right\}=\left\{y|\psi \right\}$$
3	2	eqeq1i	$$⊢\left\{x|\phi \right\}=\left\{z\right\}\leftrightarrow \left\{y|\psi \right\}=\left\{z\right\}$$
4	3	abbii	$$⊢\left\{z|\left\{x|\phi \right\}=\left\{z\right\}\right\}=\left\{z|\left\{y|\psi \right\}=\left\{z\right\}\right\}$$
5	4	unieqi	$$⊢\bigcup \left\{z|\left\{x|\phi \right\}=\left\{z\right\}\right\}=\bigcup \left\{z|\left\{y|\psi \right\}=\left\{z\right\}\right\}$$
6		df-iota	$$⊢\left(\iota x|\phi \right)=\bigcup \left\{z|\left\{x|\phi \right\}=\left\{z\right\}\right\}$$
7		df-iota	$$⊢\left(\iota y|\psi \right)=\bigcup \left\{z|\left\{y|\psi \right\}=\left\{z\right\}\right\}$$
8	5 6 7	3eqtr4i	$$⊢\left(\iota x|\phi \right)=\left(\iota y|\psi \right)$$