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.