Theorem cbvabw

Description: Rule used to change bound variables, using implicit substitution. Version of cbvab with a disjoint variable condition, which does not require ax-10 , ax-13 . (Contributed by Andrew Salmon, 11-Jul-2011) (Revised by Gino Giotto, 23-May-2024)

	Ref	Expression
Hypotheses	cbvabw.1	$$⊢Ⅎy\phi$$
	cbvabw.2	$$⊢Ⅎx\psi$$
	cbvabw.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvabw	$$⊢\left\{x|\phi \right\}=\left\{y|\psi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		cbvabw.1	$$⊢Ⅎy\phi$$
2		cbvabw.2	$$⊢Ⅎx\psi$$
3		cbvabw.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
4	1 2 3	sbievg	$$⊢\left[z/x\right]\phi \leftrightarrow \left[z/y\right]\psi$$
5		df-clab	$$⊢z\in \left\{x|\phi \right\}\leftrightarrow \left[z/x\right]\phi$$
6		df-clab	$$⊢z\in \left\{y|\psi \right\}\leftrightarrow \left[z/y\right]\psi$$
7	4 5 6	3bitr4i	$$⊢z\in \left\{x|\phi \right\}\leftrightarrow z\in \left\{y|\psi \right\}$$
8	7	eqriv	$$⊢\left\{x|\phi \right\}=\left\{y|\psi \right\}$$