std::is_same equivalent for unspecialised template types

In one project I have found a possibility to stay DRY as a lot of code except for some small parts could stay the same for template specialisations of a template. Here is a small working example what I'm currently doing to check which templated class I'm using:

template<typename T>
class A{};

template<typename T>
class B{};

template<template<class> class C>
void do_stuff()
{
    if(std::is_same<A<int>,C<int>>::value)
    {
    // Do Stuff for A
    } else if(std::is_same<B<int>,C<int>>::value)
    // Do Stuff for B
    }
}

int main()
{
    do_stuff<A>();
}

What I would like to do instead is using

std::is_same<A,C>::value

to determine the template type. Is there any function that could help me or am I missing a pattern that would work better in this case?

I see that I could do something like

template<template<class> class C, typename T>
void do_stuff();

do_stuff<A,T>();

but that seems like the wrong way to do for me.