Theorem 2ralbidva

Description: Formula-building rule for restricted universal quantifiers (deduction form). (Contributed by NM, 4-Mar-1997) Reduce dependencies on axioms. (Revised by Wolf Lammen, 9-Dec-2019)

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

Proof

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