Theorem rabeq

Description: Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003) Avoid ax-10 , ax-11 , ax-12 . (Revised by Gino Giotto, 20-Aug-2023)

	Ref	Expression
Assertion	rabeq	$$⊢A=B\to \left\{x\in A|\phi \right\}=\left\{x\in B|\phi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		eleq2	$$⊢A=B\to \left(x\in A\leftrightarrow x\in B\right)$$
2	1	anbi1d	$$⊢A=B\to \left(\left(x\in A\wedge \phi \right)\leftrightarrow \left(x\in B\wedge \phi \right)\right)$$
3	2	rabbidva2	$$⊢A=B\to \left\{x\in A|\phi \right\}=\left\{x\in B|\phi \right\}$$