Do you mean "(Ǝx. ~P(x)) ---> ~∀x. P(x)" or "Ǝx. (~P(x) ---> ~∀x. P(x))"? I personally like the convention where the scope of ∀ is as small as possible, so that "Ǝx ~P(x) ---> ~∀x P(x)" means "(Ǝx ~P(x)) ---> ~∀x P(x)," but when a quantifier is followed by a dot, then its scope extends as far as possible, so that "Ǝx. ~P(x) ---> ~∀x. P(x)" means "Ǝx (~P(x) ---> ~∀x P(x))."

Assuming you need (Ǝx ~P(x)) ---> ~∀x P(x), an informal proof is as follows. Assume Ǝx ~P(x) and break into into some x and a proof of ~P(x) using existential elimination. To prove ~∀x P(x), assume ∀x P(x). Instantiate it with x to get P(x). It gives a contradiction with ~P(x). Closing the assumption ∀x P(x), we get ~∀x P(x).

Do you need a Fitch-style or tree-like derivation?

What do you mean by Ƚ and ȽE?