Definition:Freely Substitutable

Definition
Let $\mathbf C$ be a WFF of predicate calculus.

Let $x$ be a variable in $\mathbf C$.

Let $\phi \left({y_1, \ldots, y_n}\right)$ be a term in which the variables $y_1, \ldots y_n$ occur.

Then $\phi \left({y_1, \ldots, y_n}\right)$ is freely substitutable for $x$ in $\mathbf C$ iff no free occurrence of $x$ occurs in a well-formed part of $\mathbf C$ which is of the form:
 * $( Q y_i: \mathbf B )$

where $Q$ is a quantifier, $1 \le i \le n$, and $\mathbf B$ is a WFF.

We use free for as a convenient abbreviation for freely substitutable for.

Also see

 * Confusion of Bound Variables, showing what goes wrong when substituting a variable that is not free for another.