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 GG, 14-Aug-2025)

	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		id	$$⊢x=y\to x=y$$
4	3 1	eleq12d	$$⊢x=y\to \left(x\in A\leftrightarrow y\in B\right)$$
5	4 2	anbi12d	$$⊢x=y\to \left(\left(x\in A\wedge \phi \right)\leftrightarrow \left(y\in B\wedge \psi \right)\right)$$
6	5	cbvabv	$$⊢\left\{x|\left(x\in A\wedge \phi \right)\right\}=\left\{y|\left(y\in B\wedge \psi \right)\right\}$$
7		df-rab	$$⊢\left\{x\in A|\phi \right\}=\left\{x|\left(x\in A\wedge \phi \right)\right\}$$
8		df-rab	$$⊢\left\{y\in B|\psi \right\}=\left\{y|\left(y\in B\wedge \psi \right)\right\}$$
9	6 7 8	3eqtr4i	$$⊢\left\{x\in A|\phi \right\}=\left\{y\in B|\psi \right\}$$