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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?)

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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.

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