"Intrude" faces instead of extruding I want to create a stairs-like extrusion with a face - But kind of inverted, like intruded.

The result should be something like this.


However I don't want to extrude the left face to the top, but instead to intrude the right face to the bottom. But when I do this, the following happens:


I suppose this is exactly the right thing and handled perfectly well by Blender, it's just not what I want. The outer faces still exist, while the top face is just moved to the inside.

I hope you understand what I mean, if not, please leave a comment.
Like in cegatron's answer, you should use Loopcut (CtrlR)

Once you have made a new edge loop, delete the faces with X or Delete> Faces, then fill the holes with F: