Proof of De Morgan's law

Question

Could someone prove this De'Morgan law: $$(\exists x) \neg P(x,y) \iff \neg (\forall x) P(x,y) $$

I am new at this class , never had logic's before so it's pretty hard for me to do this one from scratch.

Answer 1

In the forward direction you can reason in the following manner. Left hand side say that there exist $x$, let it be $x_0$, such that $P(x_0,y)=0$. Now what right hand side says is that $P(x,y)$ is not true for all x, which is true because we know at least one $x=x_0$ which makes it false. You can reason similarly in the reverse direction.