Is there a short five lemma for fibrations in algebraic topology (in whatever category where it would be suitable -- the topological category, the homotopy category, whatever).
By short five lemma I mean as follows. Let $E \to B$ is a fibration, with fiber $F$, and $E' \to B$ be another fibration with fiber $F'$. Suppose there are maps from $F \to F'$, $E \to E'$, and $B \to B'$ that all satisfy the obvious commutative diagram. Suppose all of the spaces are connected. If the maps from $F$ and $B$ are isomorphisms in the appropriate category, is the map from $E$ an isomorphism?