Negation of statement of particular form

Question

I have a symbolic logic statement that I'm trying to negate:

for i = 1, 2, 3,...; $\forall$x $\in$[1, n] $\forall$y $\in$[1, n] p => q

Now, my attempt of negating the above statement is:

for i = 1, 2, 3,...; $\exists$x $\in$[1, n] $\exists$y $\in$[1, n] p ^ ~q

Is my attempt correct. I am thinking it is wrong since I am thinking for i = 1, 2, 3,... showed be there exists i = 1, 2, 3,...

Also does the statement : $\forall$x $\in$[1, n] $\forall$y $\in$[1, n] p => q execute one at a time for whatever i is equal to i.e. if i = 1, then the statement: $\forall$x $\in$[1, n] $\forall$y $\in$[1, n] p => q is in a sense executed for all x and y positions before i = 2 and the same is done.

Answer 1

Let me rewrite it in a slightly different way:

$\forall i \in \mathbb{N}; \forall x \in [1,n]; \forall y \in [1,n]: p \Rightarrow q$.

And the negation of it is:

$\exists i \in \mathbb{N}; \exists x \in [1,n]; \exists y \in [1,n]: p \: \wedge (\neg q)$.

I hope this helps.