Let $A$ be a commutative ring (not necessarily noetherian).

Let $I\subseteq J\subseteq A\,$ be two finitely generated ideals.

Let us denote the completion functor by $\Lambda_K (M) = \varprojlim_n M/K^nM$.

I would like to compare the two rings:

$\Lambda_J(A)$ and $\Lambda_{J'} (\Lambda_I(A))$ where $J' = J\Lambda_I(A)$.

Are they the same? are they isomorphic?

I thought about showing that in the right hand side, $A$ is dense in the $J$-adic topology, and that it is $J$-adically complete. Is this true? and are those two facts enough to show that it is isomorphic to the left hand side?

Edit: I should mention that in my application, the two rings $\Lambda_I(A)$ and $\Lambda_J(A)$ are noetherian.