Stuck on a quantifier logic problem

Question

I've been trying to prove this to no avail..

$\vdash\exists x(Px\rightarrow\forall xPx)$

The book gives a hint.. that it might be helpful to prove the following two before tackling the main problem:

$\{\forall x\neg(Px\rightarrow\forall xPx)\}\vdash\forall xPx$

and

$\{\forall x\neg(Px\rightarrow\forall xPx)\}\vdash \neg \forall xPx$

I proved both of these but I don't know how to apply it to the main problem.

Answer 1

Since you proved from the statement $\forall xPx$ and $\neg\forall xPx$ you can conclude that the statement is wrong. Your result is just the negation of that statement.

EDIT: This is proof by contradiction. You can assume a statement and prove that you can derive a contradiction($P\wedge \neg P$) and then you know that the result is true without depending on the assumption.

Answer 2

You can just prove it semantically (using completeness theorem).

Choose arbitrary model $M$.

If $M\models\forall x P(x)$, then clearly $M$ models the statement in question.

Else there is some $m\in M$ such that $M\models\neg P(m)$. But then clearly $M\models P(m)\rightarrow \forall x P(x)$, and then $M\models\exists x(P(x)\rightarrow \forall x P(x))$.