TITLE: Homotopy pullback of a homotopy pushout is a homotopy pushout
QUESTION [15 upvotes]: Let's assume that we have a cube of spaces such that everything commutes up to homotopy.
The following holds:
    - The right square is a homotopy pushout and
    - all the squares in the middle are homotopy pullbacks.
I was told that (in Top) the left square is also a homotopy pushout then.
Do you know any source where this is stated? (Or: is this actually true?)

REPLY [17 votes]: This is Mather's second cube theorem, see Theorem 25 in 
Mather, Michael, Pull-backs in homotopy theory, Can. J. Math. 28, 225-263 (1976). ZBL0351.55005.