how is (∀v₁)Qv₂θ equivalent to Qv₂(∀v₁)θ and (∃v₁)Qv₂θ equivalent to Qv₂(∃v₁)θ? (v=variable, Q=quantifier, θ=statement)

Question

I'm reading a logic book, and i'm learning how to convert statements to prenex form, and my book says:

Replace (∀v₁)Qv₂θ with Qv₂(∀v₁)θ

Replace (∃v₁)Qv₂θ with Qv₂(∃v₁)θ

where

Q stands for an arbitrary quantifier

v stands for an arbitrary variable

θ stand for arbitrary statement or expression that result from substituting one or more variable tokens for singular terms in statement

what do these rules mean? To me, it seems like they're telling me that we can change the order of the quantifiers, but i know that if the quantifiers are different we can't do it: Are $(\exists x)(\exists y)$ and $(\forall x)(\forall y)$ equivalent to $(\exists y)(\exists x)$ and $(\forall y)(\forall x)$ respectively?

Could you guys help me please? the book doesn't give me any example.

Thank you!

Your doubts are correct; there is something missing in $\theta$. As you say; $\forall x \exists y (x < y)$ is not the same as $\exists y \forall x (x < y)$. — Mauro ALLEGRANZA, Jan 25 '20 at 13:44
Maybe the trick is in the convoluted "specification" about $\theta$: it has only one variable ? — Mauro ALLEGRANZA, Jan 25 '20 at 13:45
In addition, we have to consider that the rules for Prenex normal form transformation do not need "swapping" quantifiers. — Mauro ALLEGRANZA, Jan 25 '20 at 14:24

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