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-13 . (Contributed by Andrew Salmon, 11-Jul-2011) (Revised by Gino Giotto, 10-Jan-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	sbco2v	$$⊢\left[z/y\right]\left[y/x\right]\phi \leftrightarrow \left[z/x\right]\phi$$
5	2 3	sbiev	$$⊢\left[y/x\right]\phi \leftrightarrow \psi$$
6	5	sbbii	$$⊢\left[z/y\right]\left[y/x\right]\phi \leftrightarrow \left[z/y\right]\psi$$
7	4 6	bitr3i	$$⊢\left[z/x\right]\phi \leftrightarrow \left[z/y\right]\psi$$
8		df-clab	$$⊢z\in \left\{x|\phi \right\}\leftrightarrow \left[z/x\right]\phi$$
9		df-clab	$$⊢z\in \left\{y|\psi \right\}\leftrightarrow \left[z/y\right]\psi$$
10	7 8 9	3bitr4i	$$⊢z\in \left\{x|\phi \right\}\leftrightarrow z\in \left\{y|\psi \right\}$$
11	10	eqriv	$$⊢\left\{x|\phi \right\}=\left\{y|\psi \right\}$$