This is the proof of lemma 7.2 in a notes by Stephen Mitchell, on classifying spaces.
Essentially one step of the proof claims that:
Let $p:Y \rightarrow B \times I$ be a principal bundle. If $B$ is a CW complex, and $s:B \times \{0 \} \rightarrow Y$ is a section. Then there is an extension $s':B \times I \rightarrow Y$.
It says this follows from general property of Serre fibrations. (where the author also says should be in this set of notes. )
How is this so?