Renaming quantified variables

If $y$ is a new variable that does not occur in $\varphi$, then the following equivalences hold:

$$\begin{array}{rl}\exists x\varphi (x)& \equiv \exists y\varphi (x/y)\\ \forall x\varphi (x)& \equiv \forall y\varphi (x/y)\end{array}$$

Where $\varphi (x/y)$ is obtained from $\varphi (x)$ by replacing all free occurrences of $x$ in $\varphi (x)$ with $y$.