Short Five Lemma for Fibrations 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?  
 A: Associated to a fibration tere is always a long exact sequence in homotopy groups. So if all spaces have the homotopy type of CW complexes the usual 5 Lemma tells you that if two of the three maps are (weak) homotopy equivalences, then so is the third.
A: What do you mean by the map between the fibres $F \to F'$ ?  The projections $E \to B$ and $E' \to B'$ have lots of fibres.  Do you mean that the map $\phi \colon E \to E'$ sends every fibre of $E \to B$ to a fibre of $E' \to B'$ and on each one is an isomorphism or do you just mean it sends one fibre in that way ?   
If the former and you are interested in manifolds and isomorphisms are diffeomorphisms the result is true. A proof would be to notice that the map $E \to E'$ is a bijection because it is a bijection on fibres and the map on the base is a bijection.  Hence there is a set-theoretic inverse. If $e \in E$ then the condition of being a diffeomorphism on the fibre through $e$ and on the base will be enough to show that the derivative of this map is a bijection.  Hence the inverse function theorem shows there is a smooth local inverse which must coincide with the global set-theoretic inverse. 
Not sure about the topological category with homeomorphisms as isomorphisms.  Does that follow from mland's answer ?  Is a weak homotopy inverse which is a bijection a homeomorphism ?
