First Order Logic Statement for Proof. Manipulating Quantifiers

Question

Ok, I have the given relation: If F(x) is not true then no case satisfies G(x) and H(y,x). ((∀x ¬F(x)) ⇒¬(∀y G(y) ˄ H(y,x)))

Now, Can I possibly convert this into: (∀y G(y) ˄ H(y,x))) ⇒ ((∀x F(x)) ????

If not, the left hand side essentially has to imply: If F(x) is not true.... Mentions nothing about the For All or Existential Quantifiers. Can I take the negation outside of the Quantifier i.e. put it as (¬(∀x F(x)), because this makes the job much easier???

Answer 1

I'm not sure this is the right place but, no you can't. Moving the negation out would change the quantifier. Also, the initial formula may not be what you want: the last x is a free variable.