Blackadar's K-Theory for operator algebras has it, although the way it is done there is perhaps overkill if this is all you need. The result is generalized to local $C^*$-algebras, and they show similarity by showing the stronger property of homotopy equivalence. It is Proposition 4.6.2 on page 23 of the 2nd edition (1998). (Proposition 4.3.3 shows that homotopy equivalence is stronger.)
The stronger equivalence (but just for $C^*$-algebras) is also shown in the K-theory book by Rørdam et al., Lemma 11.2.7, with a very similar proof.