Spot the distribution: A difference between two normal CDFs

Does anyone know if this is a known distribution with known properties? It would assist me in writing a model if I knew I were working with a known family of distributions.

My pdf is constructed out of a normal pdf conditional on knowing is in some interval. For clarity: let be the normal pdf and the normal cdf, with the usual parameters . The distribution family I am trying to find the name for is the conditional pdf given that lies in some range:There are a few ways to write this integral. I include all of them in case anyone has seen any of them before:

- Use to represent the normal CDF with parameters .

Then, my conditional PDF can be written:

- , where Erf is the error function.

- A third way is to write the integral explicitly:

The pdf seems to have a number of nice properties. For example, it is symmetric and the mean is . Its variance is . Its graph looks a little like a generalized normal, and indeed it seems to tend toward the uniform as . I'm looking for guidance because I don't want to reinvent the wheel!

Many, many thanks if you know this!