Theorem r19.28zv

Description: Restricted quantifier version of Theorem 19.28 of Margaris p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 19-Aug-2004)

	Ref	Expression
Assertion	r19.28zv	$$⊢A\ne \varnothing \to \left(\forall x\in A\left(\phi \wedge \psi \right)\leftrightarrow \left(\phi \wedge \forall x\in A\psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		nfv	$$⊢Ⅎx\phi$$
2	1	r19.28z	$$⊢A\ne \varnothing \to \left(\forall x\in A\left(\phi \wedge \psi \right)\leftrightarrow \left(\phi \wedge \forall x\in A\psi \right)\right)$$