Construction of $\exists x_i (\phi)$ in first order logic.

Question

How do we construct $\exists x_i(\phi)$ where $\phi$ a formula from first order language by using:

• The logical symbols $(,),\neg,\to,\forall$
• The variable symbols $x_i$

Answer 1

There exists an x such that phi holds $\equiv$ It is not the case that for all x, phi does not hold.

ie. $\exists x_i(\phi) = \neg(\forall x_i(\neg\phi))$