Age problem

John and Bettan are friends with Noah. They wanted to know when Noah's birthday is, Noah gives list of 10 possibilities :

$\text{May 15} \quad\quad\quad\text{ May 16} \quad\quad\quad\text{ May 19}\\ \text{June 17} \quad\quad\quad\text{ June 18} \\ \text{July 14} \quad\quad\quad\text{ July 16} \\ \text{August 14}\quad\quad\quad\text{ August 15}\quad\quad\quad\text{ August 17}$

Noah tells his birthday's day and month respectively to Bettan and John.

$\text{John : I don't know when's Noah birthday, sure John doesn't knew too} \\ \text{Bettan : At first I too don't know, but I now know his birthday} \\ \text{John : I too knew Noah's birthday}$

So, when is Noah's birthday?

* If you got the answer, post a solution how. Just don't guess, apply a lot of logic *

This problem has been asked by a 14 year old student!