Theorem raleqbidv

Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007) Remove usage of ax-10 , ax-11 , and ax-12 and reduce distinct variable conditions. (Revised by Steven Nguyen, 30-Apr-2023)

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

Proof

Step	Hyp	Ref	Expression
1		raleqbidv.1	$$⊢\phi \to A=B$$
2		raleqbidv.2	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
3	1	eleq2d	$$⊢\phi \to \left(x\in A\leftrightarrow x\in B\right)$$
4	3 2	imbi12d	$$⊢\phi \to \left(\left(x\in A\to \psi \right)\leftrightarrow \left(x\in B\to \chi \right)\right)$$
5	4	ralbidv2	$$⊢\phi \to \left(\forall x\in A\psi \leftrightarrow \forall x\in B\chi \right)$$