Let $w$ be a group word with two variables $x$ and $y$.
Is the sentence $(\forall x)(\exists y)w=1$
true in every group if it is true
in every finite group?
The same question about the sentence $(\exists x)(\forall y)w=1$.

EDIT: For both questions the answers below say YES.
Here are two more complicated similar questions. Let 
$w$ be a group word with three variables $x,y,z$.
Is the sentence $(\forall xy)(\exists z)w=1$ true in
all groups if it is true in all finite groups?
The same question for the sentence
$(\exists x)(\forall yz)w=1$.