Theorem rabeqi

Description: Equality theorem for restricted class abstractions. Inference form of rabeqf . (Contributed by Glauco Siliprandi, 26-Jun-2021) Avoid ax-10 , ax-11 , ax-12 . (Revised by GG, 3-Jun-2024)

	Ref	Expression
Hypothesis	rabeqi.1	$$⊢A=B$$
Assertion	rabeqi	$$⊢\left\{x\in A|\phi \right\}=\left\{x\in B|\phi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		rabeqi.1	$$⊢A=B$$
2	1	eleq2i	$$⊢x\in A\leftrightarrow x\in B$$
3	2	anbi1i	$$⊢\left(x\in A\wedge \phi \right)\leftrightarrow \left(x\in B\wedge \phi \right)$$
4	3	rabbia2	$$⊢\left\{x\in A|\phi \right\}=\left\{x\in B|\phi \right\}$$