A catheter is transluminally advanced to a heart of a subject. Via the catheter, a prosthetic valve is advanced to the native valve. The prosthetic valve includes an upstream skirt, and a downstream skirt that has a first portion and a second portion. The first portion defines a lumen therethrough. The second portion circumscribes the first portion, and defines a plurality of anchoring arms that are coupled to the second portion. A plurality of prosthetic leaflets, within the lumen, are attached to the first portion. The upstream skirt is expanded in the atrium, and the anchoring arms are expanded in the ventricle, the anchoring arms extending radially outward and toward the upstream skirt. Subsequently, native valve tissue is squeezed between the upstream skirt and the anchoring arms by causing the anchoring arms to move with respect to the prosthetic leaflets, and toward the upstream skirt.