Let $X$ be a separable Banach space. Is this property equivalent to the approximation property?

There exists a chain $X_n$ of finite-dimensional subspaces of $X$, each being a range of some projection $P_n$, such that $\bigcup X_n$ is dense in $X$ and for every $x\in X$ we have $P_nx \to x$ as $n\to \infty$?

Does this depend on the choice of a chain of finite dimensional subspaces with dense union?