Definition:Substitution for Free Occurrence

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

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

Let $$y$$ be an individual symbol (i.e. either a variable or a parameter).

Let $$y$$ be freely substitutable for $$x$$ in $$\mathbf{C}$$

We denote:
 * $$\mathbf{C} \left({x // y}\right)$$

to be the result of replacing all free occurrences of $$x$$ in $$\mathbf{C}$$ by $$y$$.

This is referred to as the substitution of $$y$$ for free occurrence of $$x$$.

Note the insistence that $$y$$ must be freely substitutable for $$x$$.

Example
Let $$\mathbf{C}$$ be the WFF:
 * $$R \left({x}\right) \or \left({Q \left({x}\right) \implies \exists x: P \left({x, z}\right)}\right)$$.

Then $$\mathbf{C} \left({x // u}\right)$$ is the WFF:
 * $$R \left({u}\right) \or \left({Q \left({u}\right) \implies \exists x: P \left({x, z}\right)}\right)$$.

Note that the second and third occurrences of $$x$$ in $$\mathbf{C}$$ are not free but bound occurrences of $$x$$.