# Inequality with non-orthogonal projections

Suppose I have a projection $$P = A(A^T BA)^{-}A^TB$$ where $$A^{-}$$ denotes the Moore-Penrose pseudo-inverse of $$A$$ and $$B$$ is symmetric and positive definite. I am wondering, is it true that for any symmetric and positive definite matrix $$X$$, $$Y = PXP^T - X$$ is non-negative definite? If so, how would one prove this?