Order of quantifiers

The order of quantifiers matters depending on the formula. For a sequence of same-type quantifiers, the order does not matter:

$$\forall x\forall y\varphi (x,y)\equiv \forall y\forall x\varphi (x,y)$$

For a sequence of alternating quantifiers, the order matters:

$$\exists x\forall y\varphi (x,y)\not\equiv \forall y\exists x\varphi (x,y)$$