What is the property of a PL that extracting a subroutine should not change the meaning?
I believe you are looking for Morris style Contextual Equivalence. Which says that the meaning of
a term (in your case a subroutine) should not change in any context.

If, $⟦ \cdot ⟧ : term \to \mathcal{D}$, is the meaning function then
$$\forall C. ⟦C[t]⟧ = ⟦ C ⟧ \circ ⟦ t ⟧ $$
Where $C[\cdot]$ is a context with a hole in it.