In this paper we establish the link between strategy-proofness and unanimity in a domain of private good economies with single-peaked preferences. We introduce a new condition and we prove that if this new property together with the requirement of citizen sovereignty are held, a social choice function satisfies strategy-proofness if and only if it is unanimous. As an implication, we show that strategy-proofness and Maskin monotonicity become equivalent. We also give applications to implementation literature: We provide a full characterization for standard Nash implementation and partially honest Nash implementation and we determine a certain equivalence among these theories.