The idea is for these classes to use functions from each other.
That's not what you're doing: Inheritance goes only one way. mysql, user, and etc inherit mains abilities but not those of each other.
You could have mysql extend user extend etc so that at least mysql can call all the functions but that probably won't make sense, as I think they are not ancestors but siblings to each other, fulfilling a distinctly different function.
You would have to either define any shared methods in main - often the best way to go - or introduce the classes to each other so they're able to call each other's functions. You could, for example, add an instance of each needed class as parameters to the constructor:
$etc = new etc();
$mysql = new mysql($etc);
// mysql's constructor sets $this->etc
// so that it can call etc's functions using $this->etc->function()
or, more complex, using the dependency injection or singleton patterns. I asked a related question some time ago about how to deal with this in PHP and got a lot of feedback, maybe it gives you some inspiration.