Thinning a Renewal Process – Poisson Generalization

If we have a Poisson point process with rate $\lambda$ and we keep each of its point with probability $p$, we obtain another Poisson point process with rate $\lambda p$. Does this result holds for a general Renewal point process? i.e.,

If in a renewal point process with rate $\lambda$, we keep each point with probability $p$ independently, do we obtain another renewal point process?

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