If $x\equiv y\text{ mod }P^j$ for all $j$, then does it imply that $x=y$ ?

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Let $R$ be a commutative ring with identity. Let $P$ be a prime ideal of $R$. If $x\equiv y\text{ mod }P^j$ for all $j\in\mathbb{N}$, does it imply $x=y$ ?