Who was the first to prove this theorem and is there an "official" name for it?

Let $\phi:X\rightarrow Y$ be a map of H-spaces that are also CW-complexes. Assume $\phi$ induces isomorphisms on homology groups $H_{*}(-,\mathbb{Z})$. Then $\phi$ is a homotopy equivalence.

I learned about this theorem from my undergraduate topology professor who called it "Whitehead's theorem for H-spaces". The only source I have for it is V. Srinivas, Algebraic K-theory, Thm. A.53. It does not appear in any of J.H.C. Whitehead's writings, and I guess my professor called it so because of the analogy with Whitehead's theorem on weak homotopy equivalences. Does anybody know where the first proof appeared and if that name is official?