Theorem r19.28v

Description: Restricted quantifier version of one direction of 19.28 . (The other direction holds when A is nonempty, see r19.28zv .) (Contributed by NM, 2-Apr-2004) (Proof shortened by Wolf Lammen, 17-Jun-2023)

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

Proof

Step	Hyp	Ref	Expression
1		id	$$⊢\phi \to \phi$$
2	1	ralrimivw	$$⊢\phi \to \forall x\in A\phi$$
3	2	anim1i	$$⊢\left(\phi \wedge \forall x\in A\psi \right)\to \left(\forall x\in A\phi \wedge \forall x\in A\psi \right)$$
4		r19.26	$$⊢\forall x\in A\left(\phi \wedge \psi \right)\leftrightarrow \left(\forall x\in A\phi \wedge \forall x\in A\psi \right)$$
5	3 4	sylibr	$$⊢\left(\phi \wedge \forall x\in A\psi \right)\to \forall x\in A\left(\phi \wedge \psi \right)$$