Assume that we define the operator $\leadsto$ between two propositions $p$ and $q$ as follows:
$$\begin{array}{|c|c|} \hline p & q & p \leadsto q \\\hline \text{T} & \text{T} & \text{F}\\ \hline \text{T} & \text{F} & \text{T}\\ \hline \text{F} & \text{T} & \text{F}\\ \hline \text{F} & \text{F} & \text{T}\\ \hline \end{array}$$
Now suppose that we want to rewrite the logical equivalent of the statement $p \leadsto q$ using only logical conjunction, disjunction, and negation operators. How can this be done? is it possible? In general, is it possible to rewrite any logical operator in mathematical logic (such as the one we arbitrarily defined above) using only logical conjunction, disjunction, and negation operators?
It was proved by I think Sheffer and independently in Wittgenstein's Tractatus that all (!) logical connectives in a two-valued logic can be defined using only one basic operator, namely "neither A, nor B". (which is only true iff both A and B are false). We write this as $A|B$.
Here the basic procedure. First, we define negation: $\neg A := A|A$. So, "not A" is defined as "neither A nor A". Then, we can define conjunction thus: $A \land B := \neg A | \neg B$. That is, "neither not A nor not B". If this sounds confusing, check the truth table that you get. Then, we proceed to define the other common operators as is usual. (So, $A \lor B := \neg(\neg A \land \neg B)$, $A \to B := \neg A \lor B$, $A \leftrightarrow B := (A \to B) \land (B \to A)$.)
Having this in place, it's possible to define any operator. Just look at the rows in the truth table where the proposition is true, and then form a disjunction of the truth conditions you find. In your case, this would be "A and not B, or not A and not B". (See the Wikipedia entry on "disjunctive normal form".) As "not, and, or" are all defined only using "neither, nor", one can see that any logical operator can be defined using only this one basic operator.