Definition:Substitution for Free Occurrence

Definition
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:
 * $R \left({x}\right) \lor \left({Q \left({x}\right) \implies \exists x: P \left({x, z}\right)}\right)$.

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

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

Alternative Notation
Some sources use the notation $\mathbf C \left({x \gets y}\right)$ for $\mathbf C \left({x // y}\right)$.

The symbol $\gets$ can be referred to as gets, thus $\mathbf C \left({x \gets y}\right)$ is sometimes voiced as $x$ gets $y$ in $\mathbf C$.