If one wants to proof $P\vee Q$, is it sufficient to proof $\lnot P \rightarrow Q$? Because it makes intuitively more sense to me that $P\vee Q$ would be logically equivalent with $(\lnot P \rightarrow Q) \wedge (\lnot Q \rightarrow P)$.
It is in fact equivalent to $\lnot P\rightarrow Q$, which is also equivalent to $\lnot Q\rightarrow P$. So yes, it is indeed sufficient. However, in a formal logic system you should probably prove this using whatever axioms/rules of deduction you have.