Does acid and base react directly, or do they react first with water? If an acid $\ce{AH}$ reacts with a base $\ce{B}$ in aqueous solution, in textbook they write 


  $$\ce{AH + B -&gt; A- + BH+}\label{rxn:1}\tag{1}$$


But first I add acid to water and then base, so why we don't write the equations like this:


When I add acid to water:
$$\ce{AH + H2O -&gt; A- + H3O+}$$
When I add base to the acidic solution:
$$
\begin{align}
\ce{B + H2O &amp;-&gt; BH+ + OH-}\\
\ce{B + H3O+ &amp;-&gt; BH+ + H2O}
\end{align}
$$


So the equation $\eqref{rxn:1}$ isn't true because the base doesn't react directly with the acid, but with $\ce{H3O+}$ formed by the acid reaction with water. Is what I have written true?
With strong bases and acids, only $\ce{OH-}$ and $\ce{H3O+}$ react, right, because $\ce{AH}$ and $\ce{B}$ (practically) don't exist in aqueous solution. 

For weak acids and bases, that's not true, but still there the water can "transport" protons much faster between AH and B than direct exchange after diffusion. As was noted below, that's called the Grotthuss mechanism.

So yes, you are right, but the formula (1) is a valid simplification which is, for most uses, as good.