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For any semigroup S and any a, b, c, y ∈ S, in 2012 the author defined y to be a (b, c)-inverse of a if y ∈ bSy ∩ ySc, yab = b and cay = c. For given a, b, c, any such y is unique if it exists, and, by choosing b and c appropriately, one can arrange for y to become, inter alia, either the author's pseudo-inverse y = a D or (if S is a * -semigroup) the Moore-Penrose inverse y = a † . For any d ∈ S, the author proved in 1958 that da = ad implies a D d = da D whenever a D exists. In this article it is shown that, more generally, da 1 = a 2 d implies a D 2 d = da D 1 whenever a D 1 and a D 2 both exist. Still more generally, a corresponding result is proved for (b, c)-inverses; in particular, for the MoorePenrose inverse, da 1 = a 2 d and da * 1 = a * 2 d together imply a † 2 d = da † 1 . For any a 1 , a 2 ∈ S, new connections between the generalized invertibility of a 1 a 2 and a 2 a 1 are also obtained.