In order to show that this is indeed the case you will need to use so called Gauss law. It relates flux of electric field through closed surface with charge contained inside:
$$\oint \vec{E}d\vec{S} = \frac{Q}{\epsilon_{0}}$$
In general it is quite difficult to evaluate left hand side since orientation of area element $d\vec{S}$ of closed surface wrt $\vec{E}$ may vary from point to point. In some cases however it is possible to define so called Gaussian surface. It's given by the orienting $d\vec{S}$ perpendicular to $\vec{E}$(or collinear to it). For instance if you want to apply Gauss law to evaluate electric field of point charge you should pick sphere with center placed at the position of point charge.
Now we need to understand how to pick Gaussian surface for infinite plane. It is useful to start from following toy model. We place point charges uniformly along the surface. Due to symmetry of construction one can observe that electric field of nearest charges will cancel each other in all directions apart from the one exactly perpendicular to the charged surface.(Blue lines will stay uncanceled. Vertical parts of purple lines will cancel corresponding parts of nearest green ones.

You can see that in order for this cancellation to take place you need to have
1) uniformly charged plane
2) plane to be large(there won't be cancelation near the boundary of plane)
Now if we go to continuous limit we arrive to the conclusion that electric field is perpendicular to the surface of plane.(Note that the same thing will take place from the opposite side of the plane). Since we know orientation of electric field we can easily define corresponding Gaussian surface:

Now Gauss law will give us:
$$E\cdot 2 A = \frac{Q}{\epsilon_0} = \frac{\sigma A}{\epsilon_0}$$
$$E = \frac{\sigma}{2 \epsilon_0}$$
here $\sigma$ is surface charge density, $A$ - surface area of top cylinder.
So you don't really need to have two planes in order to have constant electric field.
If you take two identical infinite planes with opposite charges electric field inside will double and will cancel outside the planes.