If an acid $\ce{AH}$ reacts with a base $\ce{B}$ in aqueous solution, in textbook they write
$$\ce{AH + B -> A- + BH+}\label{rxn:1}\tag{1}$$
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: $$\ce{AH + H2O -> A- + H3O+}$$
When I add base to the acidic solution: $$ \begin{align} \ce{B + H2O &-> BH+ + OH-}\\ \ce{B + H3O+ &-> BH+ + H2O} \end{align} $$
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?