Theorem bj-rabeqbid

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

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

Proof

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