transform orthonormal coordinate system to another

Question

I have one orthonormal coordinate system ABC that it's origin is the point p0.

I would like to transform it to another orthonormal coordinate system A'B'C', that it's origin is p1.

I know how to transform from the standard orthonormal coordinate system XYZ, to any other coordinate system, that it's origin is (0,0,0). So I thought that I should transform the ABC coordinate system, to the standard one, and then to the new requested one A'B'C'.

Wether it's the simplest option or not, I'd like to know what am I missing please, and please feel free to offer better, more simple solutions.

1. I have first translated the ABC coordinate system by -p0, so both ABC and XYZ, share the same origin (0,0,0).

2. Now, using the base change matrix, to the standard base, which is the identity matrix (I), I transform the ABC system to XYZ.

3. Now I shall transform the system, to the A'B'C' system by another base change matrix which is:

   | A'x A'y A'z |
   | B'x B'y B'z |
   | C'x C'y C'z |
   

4. Translate the system by p1, so the system's origin will be p1.

Should I "cancel" the transformation to the standard system? or that's it?

Thank you.