(1) $\forall x P(x) \wedge \forall x Q(x)$
(2) $\forall x (P(x) \wedge Q(x)) $
(3) $\forall y (\forall x P(x) \wedge Q(y))$
(4) $ \forall y \forall x(P(x)\wedge Q(y))$
I'm sure that (1) and (2) are equivalance. I think (3) and (4) are the same as (1) and (2) as $\forall x (B \wedge P(x)) \Leftrightarrow B \wedge \forall x(P(x)),$ but was not sure.
Anyone have ideas about this?
Thanks!