Let $\Sigma$ be a theory and $\psi$ a sentence. I'm familiar with the notion of $\Sigma \models \psi$, however, lately, I've seen some authors using this notation when $\psi$ is a formula with free variables. For example, in Marker's "Model Theory: An Introduction" he says:
We say that a theory $T$ has quantifier elimination if for every formula $\phi$ there is a quantifier-free formula $\psi$ such that $$T\models \phi \leftrightarrow\psi$$
Here, the formula $\phi \leftrightarrow\psi$ has free variables and this confused me a lot. Nowhere in the book was I able to find a definition of $\Sigma \models \phi$ when $\phi$ has free variables.
So I'm here to ask what is the most common way of defining $\Sigma \models \phi$ where $\phi$ may have free variables.
