In Mitchell's book "Theory of Categories", Corollary I.16.8 (page 24) states that the following holds in any exact category:

Let $$ 0 \to A \to B \to C \to 0 $$ $$ 0 \to B^' \to B \to B^{''} \to 0 $$ be short exact sequences. Then $B^' \to B \to C$ is epi iff $A \to B \to B^{''}$ is epi.

It seems to me that Mitchell's proof requires the existence of pushouts and pullbacks. Therefore I wonder if the corollary actually is true for any exact category. Can someone acknowledge this corollary ?

The reason why I think Mitchell's proof requires pushouts and pullbacks is as follows: In a first step the two short exact sequences are embedded crosswise into a commutative diagram with three short exact columns and three short exact rows. But according to Proposition I.16.5 such a diagram only exists if some of its squares are a pushout or a pullback.