Theorem rabeqbidva

Description: Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017) Remove DV conditions. (Revised by GG, 1-Sep-2025)

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

Proof

Step	Hyp	Ref	Expression
1		rabeqbidva.1	$$⊢\phi \to A=B$$
2		rabeqbidva.2	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
3	2	rabbidva	$$⊢\phi \to \left\{x\in A|\psi \right\}=\left\{x\in A|\chi \right\}$$
4	1	eleq2d	$$⊢\phi \to \left(x\in A\leftrightarrow x\in B\right)$$
5	4	anbi1d	$$⊢\phi \to \left(\left(x\in A\wedge \chi \right)\leftrightarrow \left(x\in B\wedge \chi \right)\right)$$
6	5	rabbidva2	$$⊢\phi \to \left\{x\in A|\chi \right\}=\left\{x\in B|\chi \right\}$$
7	3 6	eqtrd	$$⊢\phi \to \left\{x\in A|\psi \right\}=\left\{x\in B|\chi \right\}$$