Theorem raleqbidva

Description: Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017)

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

Proof

Step	Hyp	Ref	Expression
1		raleqbidva.1	$$⊢\phi \to A=B$$
2		raleqbidva.2	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
3	2	ralbidva	$$⊢\phi \to \left(\forall x\in A\psi \leftrightarrow \forall x\in A\chi \right)$$
4	1	raleqdv	$$⊢\phi \to \left(\forall x\in A\chi \leftrightarrow \forall x\in B\chi \right)$$
5	3 4	bitrd	$$⊢\phi \to \left(\forall x\in A\psi \leftrightarrow \forall x\in B\chi \right)$$