Linear programming: reduce a contstraint with simple minimun

I have an almost linear programme. However one of the constraints has a form $ z = min(x,y)$ (all the other things are linear in the model). Is there a way to substitute this with something (or introduce additional variables) to turn this into a linear programme?

In other words, I have the problem that looks like the following: $ $ \mathbf c’ \mathbf x \to \min, $ $ s.t. $ $ A \mathbf x = \mathbf b,\quad x_1 = \min(x_2, x_3). $ $