Theorem rabeqbidv

Description: Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009)

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

Proof

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