Theorem 2ralbidv

Description: Formula-building rule for restricted universal quantifiers (deduction form). (Contributed by NM, 28-Jan-2006) (Revised by Szymon Jaroszewicz, 16-Mar-2007)

	Ref	Expression
Hypothesis	2ralbidv.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	2ralbidv	$$⊢\phi \to \left(\forall x\in A\forall y\in B\psi \leftrightarrow \forall x\in A\forall y\in B\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		2ralbidv.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2	1	ralbidv	$$⊢\phi \to \left(\forall y\in B\psi \leftrightarrow \forall y\in B\chi \right)$$
3	2	ralbidv	$$⊢\phi \to \left(\forall x\in A\forall y\in B\psi \leftrightarrow \forall x\in A\forall y\in B\chi \right)$$