Yes, this is a general rule of rings. Given a homomorphism
ϕ:R/I→S
with kerϕ=J, then
R/(I+J)≅S/J.

This can be shown by considering the following two short exact sequences:

    0→J→R→R/J→0
    0→J→S→S/J→0

The first short exact sequence shows that R/J is isomorphic to R modulo J. The second short exact sequence shows that S/J is isomorphic to S modulo J. Since the map
ϕ:R/I→S
satisfies kerϕ=J, it follows that
R/I≅R/J
and
S≅S/J.

Thus, by the isomorphism theorems, we have that
R/(I+J)≅(R/J)/(I/J)≅S/J.

I hope this helps to clarify the relationship between these rings and the role of the homomorphism
ϕ:R/I→S