Erios got it wrong, you had it right (you need to assign the result of the query to the query_run variable, not check them against each other).
Your code should look something like this:
echo 'You entered the if';
$query = "SELECT id FROM users WHERE username='$username' AND password='$password_hash'";
echo 'You set the query.';
$query_run = mysql_query($query);
echo 'You got a query response.';
I believe that should work. But it's possible that you have to check query_run in another way (like $result = mysql_store_result($query_run)