Is it true that all alkalis are bases, but not vice versa?

According to this statement all the alkalies are bases, but all the bases are not alkalies. This is true.

A base is defined as a chemical that neutralizes acids. In such a neutralization reaction, a base reacts with an acid to form salt and water. The generic chemical reaction for this neutralization is:

base + acid -> salt + water

Alkalies are also included in this definition.

On the other hand, an alkali is defined as a water soluble base. Another definition is that an alkali is a chemical that dissolves in water to form a solution that will have a pH of more than 7. Thus, alkalies are all bases, while all bases may not be alkalies. 

An example is copper oxide, which is insoluble in water and hence is not an alkali. However, this is a base.

Hope this helps.