I want to prove the following.
Let $X$ be a topological space and let $A\subset X$. If $(X,A)$ has the Homotopy Extension Property and $f,g:A\rightarrow Y$ are homotopic, then $X\cup_f Y$ is homotopy equivalent to $X\cup_g Y$.
There are a few ways to go about this. The first idea is to construct explicitly the maps whose compositions are homotopic to the respective identities, but I can’t really see any way to construct these.
The other route is to show these spaces are deformation retracts of the same space. Since $X$ is homeomorphic to $X\times\{0\}$, it seems like $(X\times[0,1])\cup_h Y$ is a good candidate, but I can’t really take this idea further either.