Considering U the universal set, which contains the elements 0, 1, 2..., does $\forall x \exists y P(x, y) $ mean 

$\exists yP(0,y) \wedge \exists yP(1,y) \wedge \exists yP(2,y)...$ 

or 

$\forall x[x \in U \rightarrow \exists yP(x,y)]=$  

$[0 \in U \rightarrow \exists yP(0,y)] \wedge [1 \in U \rightarrow \exists yP(1,y)] \wedge [2 \in U \rightarrow \exists yP(2,y)]...$ ?