Suppose we are told that

$$\neg\, (P \Rightarrow Q)$$    is equivalent to $$P \, \& \, \neg\, Q\,. \hspace{2cm} (*)$$

From this we suspect that we can express $P \Rightarrow Q$ by using conjunction (&), disjunction ($\vee$) or negation ($\neg$) only.

First observe that $P \Rightarrow Q$ is equivalent to

$$\neg\, (\neg \,P \Rightarrow Q)\,,$$

therefore from what we are given in (*), $P \Rightarrow Q$ is equivalent to

$$\neg\, (P \,\&\, \neg\, Q)$$

We know that given two statements $R$ and $S$ (we use different letters to distinguish these statements from the $P$ and $Q$ above) by de Morgan's Law about the negation of conjunctions,

$$\neg\,(R \,\&\,S)$$    is equivalent to $$\neg\, R\, \vee\, \neg \,S.$$

Therefore taking $P$ as the first statement $R$ and $\neg\, Q$ as the second statement $S$ in de Morgan's Law we obtain

$$\neg\, (P\, \& \,\neg Q)$$    is equivalent to $$\neg\, P \vee \neg\, (\neg\, Q).$$

Since $P \Rightarrow Q$ is equivalent to $\neg \,(P\, \&\, \neg\, Q)$ and the latter is equivalent to $\neg\, P\, \vee\, \neg \,(\neg\, Q)$ we can say that

$$P \Rightarrow Q$$    is equivalent to $$\neg\, P \vee \neg\, (\neg\, Q).$$

The expression on the right hand side can be simplified by observing that $\neg\, (\neg\, Q)$ is equivalent to $Q$. So as a result

$$P \Rightarrow Q$$    is equivalent to $$\neg \,P \vee Q.$$