On a Geometric Interpretation of the Local Criterion for Flatness

On a Geometric Interpretation of the Local Criterion for Flatness

The local criterion for flatness goes this way:
Let $\phi : (A,m)\rightarrow (B,m')$ be a local morphism of local
Noetherian rings, and $M$ a finitely generated $B$-module. If $x\in m$ is
a non zero-divisor on $M$ then $M$ is flat over $A$ iff $M/xM$ is flat
over $A/xA$.
One usual geometric interpretation (see for instance Eisenbud, Commutative
Algebra, 6.4) is :
If we have a morphism of affine varieties $X\rightarrow Y$ over
$\mathbb{A}^1$ such that the maps to $\mathbb{A}^1$ are flat and dominant,
for any point $p$ in $\mathbb{A}^1$ choose a point $p'$ in $Y$ above $p$
and a point $p''$ in $X$ above $p'$. If the map of fibers
$X_{p}\rightarrow Y_{p}$ is flat in a neighborhood of $p''$ in $X_{p}$,
then the map $X\rightarrow Y$ is also flat in a neighborhood of $p''$ in
$X$.
I fail to see the obviousness of this interpretation: I mean that using
the local criterion for flatness I see how I can get the flatness of the
rings localized at maximal ideals $p'$ and $p''$ respectively coming from
the flatness on the fibers, but how to extend it to a neighborhood of each
points ?
I posted a similar question on MO but there one rightly explained to me
that if a map is flat localized at one point it is flat on an open
neighborhood of this point, using a well known result about openness of
flat maps from EGA. But here considering the level of Eisenbud chapter 6,
or in other places in literature where this interpretation is given such a
use of this result is never mentioned. I fail to see how to get the
interpretation without using it. Can somebody enlighten me here ?