How can I implement double dispatch when I don't know all the classes in advance?

I've got a base class with (potentially) a lot of subclasses, and I would like to be able to compare any two objects of the base class for equality. I am trying to do this without invoking the blasphemous typeid keyword.

#include <iostream>

struct Base {

    virtual bool operator==(const Base& rhs) const
        {return rhs.equalityBounce(this);}

    virtual bool equalityBounce(const Base* lhs) const = 0;
    virtual bool equalityCheck(const Base* lhs) const = 0;
};

struct A : public Base {

    A(int eh) : a(eh) {}
    int a;

    virtual bool equalityBounce(const Base* lhs) const{
        return lhs->equalityCheck(this);
    }

    virtual bool equalityCheck(const Base* rhs) const {return false;}
    virtual bool equalityCheck(const A* rhs) const {return a == rhs->a;}
};


int main() {

    Base *w = new A(1), *x = new A(2);

    std::cout << (*w == *w) << "\n";
    std::cout << (*w == *x) << "\n";
}

I understand that the code as written is failing because lhs in equalityBounce() is a Base*, so it doesn't even know about the version of equalityCheck() that takes an A*. But I don't know what to do about it.