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???
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.