Sum of random variables started after a stopping time

Question

Suppose we have a stopping time $1\leq T\leq N$ with respect to the filtration of independent random variables $\{X_n\}_{n=1}^N$. Consider $S_T^N=\sum_{n=T+1}^N X_n$. Is it correct to state that $\mathbb{P}(S_T=1|T=n)=\mathbb{P}(S_n=1)$? My reasoning was the following: $\mathbb{P}(S_T=1|T=n)= \mathbb{P}(S_n=1,T=n)/\mathbb{P}(T=n)=\mathbb{P}(S_n=1)\mathbb{P}(T=n)/\mathbb{P}(T=n)=\mathbb{P}(S_n=1)$. My reasoning is the following. I think it is true because being a stopping time such that $\{T=n\}\in\sigma(X_1,\ldots,X_n)$ should imply that, being $S_n$ independent of $X_1,\ldots,X_n$, $\{S_n=1\}$ is independent of $\{T=n\}$.