Theorem r19.27v

Description: Restricted quantitifer version of one direction of 19.27 . (Assuming F/_ x A , the other direction holds when A is nonempty, see r19.27zv .) (Contributed by NM, 3-Jun-2004) (Proof shortened by Andrew Salmon, 30-May-2011) (Proof shortened by Wolf Lammen, 17-Jun-2023)

	Ref	Expression
Assertion	r19.27v	$$⊢\left(\forall x\in A\phi \wedge \psi \right)\to \forall x\in A\left(\phi \wedge \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		id	$$⊢\psi \to \psi$$
2	1	ralrimivw	$$⊢\psi \to \forall x\in A\psi$$
3	2	anim2i	$$⊢\left(\forall x\in A\phi \wedge \psi \right)\to \left(\forall x\in A\phi \wedge \forall x\in A\psi \right)$$
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	$$⊢\left(\forall x\in A\phi \wedge \psi \right)\to \forall x\in A\left(\phi \wedge \psi \right)$$