Theorem ralbidv2

Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Apr-1997)

	Ref	Expression
Hypothesis	ralbidv2.1	$$⊢\phi \to \left(\left(x\in A\to \psi \right)\leftrightarrow \left(x\in B\to \chi \right)\right)$$
Assertion	ralbidv2	$$⊢\phi \to \left(\forall x\in A\psi \leftrightarrow \forall x\in B\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ralbidv2.1	$$⊢\phi \to \left(\left(x\in A\to \psi \right)\leftrightarrow \left(x\in B\to \chi \right)\right)$$
2	1	albidv	$$⊢\phi \to \left(\forall x\left(x\in A\to \psi \right)\leftrightarrow \forall x\left(x\in B\to \chi \right)\right)$$
3		df-ral	$$⊢\forall x\in A\psi \leftrightarrow \forall x\left(x\in A\to \psi \right)$$
4		df-ral	$$⊢\forall x\in B\chi \leftrightarrow \forall x\left(x\in B\to \chi \right)$$
5	2 3 4	3bitr4g	$$⊢\phi \to \left(\forall x\in A\psi \leftrightarrow \forall x\in B\chi \right)$$