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This (for real-valued rather than complex-valued functions) was in Stone's original paper that proved the Stone-Weierstrass theorem, as Theorem 84. The statement is a bit funny, since he defines the equivalence relation $x\sim y$ not in the obvious way but as "$x$ is in the intersection of all sets $f^{-1}(U)$ such that $f\in\mathcal{A}$, $U\subset\mathbb{R}$ is an open interval, and $y\in f^{-1}(U)$" (this definition appears in the statement of Theorem 81).