Theorem raleqbidvv

Description: Version of raleqbidv with additional disjoint variable conditions, not requiring ax-8 nor df-clel . (Contributed by BJ, 22-Sep-2024)

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

Proof

Step	Hyp	Ref	Expression
1		raleqbidvv.1	$$⊢\phi \to A=B$$
2		raleqbidvv.2	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
3	2	adantr	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
4	1 3	raleqbidva	$$⊢\phi \to \left(\forall x\in A\psi \leftrightarrow \forall x\in B\chi \right)$$