Predicate Calculus From existential to universal quantifiers

Question

I'm working on a proof that I cannot manage to solve.

I have as hipothesis

$(\exists x) Fx \implies Ga $

And need to derive

$ (\forall x) (Fx \implies Ga) $

Im pretty confused since I know that the equivalences of the negation of a quantifiers have the bounded variable negated like this:

¬∃x φ is the same as ∀x ¬φ

I cannot see how an existential quantifiers can be used to generate a universal quantifiers with the same propositon.

Can anyone help me understand this deduction?

Answer 1

Your hypothesis is: $$\color{blue}{\big(}(\exists x)Fx\color{blue}{\big)} \to Ga$$

Use Implication Equivalence $A\to B ~\equiv~ \neg A\vee B$ $$\neg\color{blue}{\big(}(\exists x)Fx\color{blue}{\big)}~\vee~Ga$$

Now use Dual Negation, Distribution, then Implication Equivalence again.

$${\big(}(\forall x)\neg Fx{\big)}~\vee~Ga\\{\big(}(\forall x)(\neg Fx \vee Ga){\big)}\\{\big(}(\forall x)(Fx \to Ga){\big)}$$