$ \exists x (Fx \rightarrow \forall x Fx) $
I am trying to construct a derivation for this problem which can be found in the UCLA logic software program under chapter 3 of the derivation section. Up to this point in quantifier logic, anytime I had to prove an existential statement, there were either premises available to help derive the conclusion or the conclusion itself was an open conditional so I was able to assume the existential and just focus on proving the consequent.
However, in this case, it is a closed statement that has a conditional in it so I cannot assume the antecedent, nor is there a premise. lastly, i cannot apply quantifier negation to frame it as a universal statement. I am just completely stumped on this so any tips or suggestions or maybe even reference to resource will be helpful.
my derivation stops after the second line.
Show $ \exists x (Fx \rightarrow \forall x Fx) $
$ \space\space\space\space \neg \exists x (Fx \rightarrow \forall x Fx) \space\space\space assume \space indirect $
If $\forall x Fx$, then any element works as a witness to prove the desired statement. On the other hand, if $\lnot\forall x Fx$, then there is an element $m$ such that $\lnot Fm$, and this element works as a witness.