In his paper "Mukai flops and derived categories", Namikawa reduces a general Mukai flop of a smooth projective $2n$-dimensional variety $Z$ along a subvariety $W\cong \mathbb P^n$ with $N_{W/Z}\cong \Omega^1_{\mathbb P^n}$ to another smooth projective variety $Z^+$ with subvariety $W^+$, satisfying the above properties and obtained by blowing up $Z$ along $W$ and then blowing down the exceptional divisor in the other direction, to the simpler case where $X,X^+$ are $\mathbb P^n$ bundles over $\mathbb P^n$ (given by $\mathbb P(\Omega^1_{\mathbb P^n}\oplus \mathcal O_{\mathbb P^n})$) with the projective space subvarieties $Y,Y^+$ coming from the sections of these bundles corresponding to the quotient line bundle $\mathcal O_{\mathbb P^n}$ where the "plus" versions are obtained again by blowing up and blowing down in the other direction.

He proves the simpler case directly, and then reduces the above case to this one by noting that the formal completion of $X$ along $Y$, $X_Y$, is naturally isomorphic to $Z_W$, the formal completion of $Z$ along $W$, and likewise for the "plus" versions.

I was wondering if these identifications only relied on the fact that the normal bundles and underlying variety completed along were the same in each case. If so, can someone point me to a general reference for this technique? If not, what is the proof of these identifications, since no mention of a proof is given there?