I am confused about the following: I read yesterday that for a formula $\phi(x_1,\ldots,x_n)$ in a first order language $\mathcal{L}$ and an $\mathcal{L}$-structure $\mathcal{A}$, $\mathcal{A} \models \phi(x_1,\ldots,x_n)$ iff $\mathcal{A} \models \forall x_1 \ldots \forall x_n \phi(x_1,\ldots,x_n)$. This seems perfectly fine to me; if I understand it right, it's like saying something like $x^2 \geq 0$ is true iff $\forall x (x^2 \geq 0)$ is true.

Now consider a predicate $Q(x)$; then according to the above $\models Q(x)$ iff $\models \forall x Q(x)$. Isn't this equivalent to $\models Q(x) \leftrightarrow \forall x Q(x)$? However, $\not \models Q(x) \rightarrow \forall x Q(x)$; take for example $\mathcal{A} = (A, Q^{\mathcal{A}})$, where $A=\{a,b\}$, $Q^{\mathcal{A}}=\{a\}$ and $w: \mathrm{Var} \rightarrow A$, $w(x) = a$. What am I doing wrong?