Unitaries permuting two orthogonal projections

Let P and Q be two orthogonal projections on a separable Hilbert space, H. Wang, Du and Dou proved that there exists a unitary, U, with UPU−1=Q, UQU−1=P if and only if dim⁡(ker⁡P∩ker⁡(1−Q))=dim⁡(ker⁡Q∩ker⁡(1−P)) (both may be infinite). We provide a new proof using the supersymmetric machinery of Avron, Seiler and Simon.