Definition:Freely Substitutable

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

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

The symbol $y$ 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:
 * $Q y: \mathbf B$

where $Q$ is a quantifier and $\mathbf B$ is a WFF.

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

Example
Take the WFF:
 * $\forall x: \exists y: x < y$.

Suppose we wished to substitute $y$ for $x$.

If we paid no heed to whether $y$ were free for $x$, we would obtain:
 * $\forall y: \exists y: y < y$.

This is plainly false for the natural numbers, but $\forall x: \exists y: x < y$ is true (just take $y = x + 1$).

This problem is called confusion of bound variables.