Theorem cbvrabv2w

Description: A more general version of cbvrabv . Version of cbvrabv2 with a disjoint variable condition, which does not require ax-13 . (Contributed by Glauco Siliprandi, 23-Oct-2021) (Revised by Gino Giotto, 16-Apr-2024)

	Ref	Expression
Hypotheses	cbvrabv2w.1	$$⊢x=y\to A=B$$
	cbvrabv2w.2	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvrabv2w	$$⊢\left\{x\in A|\phi \right\}=\left\{y\in B|\psi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		cbvrabv2w.1	$$⊢x=y\to A=B$$
2		cbvrabv2w.2	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
3		nfcv	$$⊢\underset{\_}{Ⅎ}yA$$
4		nfcv	$$⊢\underset{\_}{Ⅎ}xB$$
5		nfv	$$⊢Ⅎy\phi$$
6		nfv	$$⊢Ⅎx\psi$$
7	3 4 5 6 1 2	cbvrabcsfw	$$⊢\left\{x\in A|\phi \right\}=\left\{y\in B|\psi \right\}$$