Theorem rabeqbida

Description: Version of rabeqbidva with two disjoint variable conditions removed and the third replaced by a nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019)

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

Proof

Step	Hyp	Ref	Expression
1		rabeqbida.nf	$$⊢Ⅎx\phi$$
2		rabeqbida.1	$$⊢\phi \to A=B$$
3		rabeqbida.2	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
4	1 3	rabbida	$$⊢\phi \to \left\{x\in A|\psi \right\}=\left\{x\in A|\chi \right\}$$
5	1 2	rabeqd	$$⊢\phi \to \left\{x\in A|\chi \right\}=\left\{x\in B|\chi \right\}$$
6	4 5	eqtrd	$$⊢\phi \to \left\{x\in A|\psi \right\}=\left\{x\in B|\chi \right\}$$