I have to prove if the following statement is true or false $$\forall x : (P(x) \lor Q(x)) \Leftrightarrow \forall x : P(x) \lor \forall x : Q(x)$$
I understand the first statement as, for every $x$, at least one of the functions $P$, $Q$ is true. For the second statement, at least one of the functions $P$, $Q$ is true for every $x$. Is this correct? When yes, how could I prove this? Using the distributive law will clearly render me the wrong result.