Does adding non-variable quantifiers to a proposition change the proposition?

Question

Let:

$A,B,C$ be non-empty sets.

$P(z)$ is a one variable proposition.

Then are the following propositions equivalent?

$\forall\ z \in C, P(z)$

$\forall\ x \in A, [\forall\ z \in C, P(z)] $

$\exists\ x \in A, [\forall\ z \in C, P(z)] $

$\forall\ x \in A, \forall\ y \in B, [\forall\ z \in C, P(z)] $

$\forall\ x \in A, \exists\ y \in B, [\forall\ z \in C, P(z)] $

$\exists\ x \in A, \forall\ y \in B, [\forall\ z \in C, P(z)] $

$\exists\ x \in A, \exists\ y \in B, [\forall\ z \in C, P(z)] $

If yes, why?