Contrapositive of the statement involving "for every" and "there exists"

Question

I have a statement

(∃x.(P(x) -> (∀y.P(y))))

I am trying to formulate and understand the contrapositive of the formula.

(∃x.(¬∀y.P(y) -> ¬P(x))))

This is what I got.

Is this the correct contrapositive of the original formula?

Is this contrapositive interpreted as "For every y, not P(y) implies not P(x) for some x" or "Not every y holds P(y) and this implies not P(x) for some x"?

Thanks a ton.

Answer 1

The negation of "for all y, P(y)" Is "there exists a y, such that not P(y)"

Making this correction should clarify your interpretation of the contrapositive statement.

Answer 2

the contrapositive be described as follows: $(\forall x.(\exists y \neg P(y)\Rightarrow \neg P(x))$