$$\ce{NH3 + HCl -> NH4Cl}$$
The above equation is the reaction of ammonia with hydrochloric acid (all substances are aqueous). I understand that ammonia counts as a base because it can react with water to give hydroxide ions, and $\ce{HCl}$ is an acid. By logic, this should be an 'acid-base' reaction and this link says that it is indeed an acid-base reaction but how is this so if no water is produced?
I am learning acid-base theory from the beginning and can understand why this is an acid base reaction from the perspective of the Brønsted-Lowry theory and the Lewis theory, but I am now attempting to understand the Arrhenius theory, so please explain to me, in terms of the Arrhenius theory, why this is an acid-base reaction and a 'neutralistaion' if there is no water produced and no hydroxide ions which take place in the reaction what so ever?