Falsifiable first order logic formulas

Question

My text book claims that the formula: $$\forall x (Px \lor Qx) \rightarrow (\forall x Px \lor \forall x Qx)$$ is falsifiable.

But to me it seems like a tautology / valid formula.

Is my reasoning faulted, or a typo in the book?

Answer 1

As Flan and the other commenting points out the book is right and my reasoning false. I had misunderstood the scope of the bound variables.

Take $Px$ to be $=0$, $$ to be $x=1$ and $\forall x$ to range on {0, 1}. Then $\forall x(Px \land Qx)$ is true, while $\forall x Px$ and $\forall x Qx$ can be false for $x$es being 0.

Thanks for the great comments below the answer. Think they can be useful for many people with a underdeveloped understanding of scope of variables in first order logic...