Theorem rabeqbidvaOLD

Description: Obsolete version of rabeqbidva as of 1-Sep-2025. (Contributed by Mario Carneiro, 26-Jan-2017) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		rabeqbidvaOLD.1	$$⊢\phi \to A=B$$
2		rabeqbidvaOLD.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	rabeqdv	$$⊢\phi \to \left\{x\in A|\chi \right\}=\left\{x\in B|\chi \right\}$$
5	3 4	eqtrd	$$⊢\phi \to \left\{x\in A|\psi \right\}=\left\{x\in B|\chi \right\}$$