Vacuous quantification

Vacuous quantification is when a variable $x$ has no occurrence in a formula $\varphi$, hence then binding $x$ with a quantifier in $\varphi$ has no effect. Or when the variable $x$ is already bound by another quantifier in a formula $\varphi$, hence binding $x$ again has no effect.

$$\begin{array}{rl}\forall x\forall y(P(y)\to Q(y))& \equiv \forall y(P(y)\to Q(y))\\ & \equiv \exists x\forall y(P(y)\to Q(y))\\ & \\ \forall x(\forall x(P(x)\to Q(x)))& \equiv \forall x(P(x)\to Q(x))\\ \exists x(\forall (P(x)\to Q(x)))& \equiv \forall x(P(x)\to Q(x))\end{array}$$

title: not sure if the last example is correct, check if it should $\equiv \exists$