I am studying a construction of the binary Golay code using the extended binary Golay code $\mathcal{G}$. It is well known that the extended Golay code is a $[24,2^{12},8]_2$-code and can be constructed from the generating matrix $ {I} \choose {A} $ where $I$ is the $12 \times 12$ identity matrix and $A$ is the matrix of anti-adjacency of the set of vertices of an icosahedron. I studied some usual results about the extended binary Golay code. For instance: $A$ is orthogonal, or $\forall \; x \in \mathcal{G}: w(x) \in \{0,8,12,16,24\}$ where $w(x)$ denotes the weight of $x$, ...
Now it is stated that it is possible to construct a perfect $[23,2^{12},7]_2$-code with the following construction: consider the extended Golay code $\mathcal{G}$, pick $i \in \{1, \ldots, 24\}$ and set $\mathcal{G}_i := \{x_1 \ldots \hat{x}_i \ldots x_{24} \mid x_1 \ldots x_{24} \in \mathcal{G} \}$ Then $\mathcal{G}_i$ is a perfect $[23,2^{12},7]_2$-code.
Okay, now checking the perfection is not a problem since it suffices to check the Hamming bound (if we assume that $\mathcal{G}_i$ is indeed a $[23,2^{12},7]_2$-code). However why isn't it possible to choose $i \in \{1, \ldots , 24\}$ in such a clever way that $\mathcal{G}_i$ has minimal distance 8 instead of 7? I know from all the informations on the internet about Golay code that it is not possible, however I would like to see a direct proof of that, i.e. that does not involves uniqueness of the Golay code for instance. If I rephrase my question, it becomes: if we select two words at distance $8$, we can choose $i \in \{1, \ldots, 24\}$ such that the $i^{th}$ letter differs and delete it in every word so that minimal distance becomes indeed 7. Why is it always the case?