I am interested in a reference for the proof of the following theorem for $A,X$ being CGWH topological spaces.

Let $A\subset X$ be a closed subspace, such that there exists a continuous $\phi : X \rightarrow I$ with $A=\phi^{-1}(0)$. Let $U=\phi^{-1}([0,1))$.

Suppose there exists a homotopy $h:U \times I \rightarrow X$, relative to $A$ (i.e. $g(a,t)=a$ for all $a\in A$ and $t \in I$), satisfying moreover the conditions:

- $g(x,0)=x$ for all $x\in U$
- $g(x,1) \in A$ for all $x\in U$
Then the inclusion $A\subset X$ is a cofibration.

I am also interested in any proof working a priori just for locally compact spaces, in which the fact of being locally compact appears only as a justification of properties of map spaces, as the proof may then work for CGWH spaces, too.

I would be especially glad to see a proof without any sort of "hand-waving" or omissions of form "it's easy to see the map is continuous". I have already managed to be discouraged to look for justifications of such claims by myself:

In Postnikov's "Lectures in algebraic topology: Elements of homotopy theory" (1984, in Russian) I found a proof of this statement, but it happened to have a flaw (some map claimed to be continuous, without any argument to it, wasn't necessarily such) and after some little modification it still wasn't clear to me it would work without additional assumption of $X$ being locally compact (but with this assumtpion the modification seems to make the reasoning valid).

I have looked up another proof, now not sure of where to find it again, which suffered from the same kind of omission (an unproven claim that some map is continuous, which happened to be false).