Every conjugate of Sylow permutable is Sylow permutable.

Question

I am working on a proof for the statement: 'If $H$ is a Sylow permutable subgroup of a finite group $G$, then every conjugate of $H$ is Sylow permutable in $G$.
' My current proof is based on the observation that if $K=H^g$ and $P$ is any Sylow subgroup of $G$, then $KP=H^gP=(HP^{g^{-1}})^g$, which is a subgroup of $G$. Consequently, $KP=PK$, and $K$ is Sylow permutable permute in $G$.
I would like to ask if my proof is formulated properly and logically sound. Any suggestions to make it clearer or easier to follow are also appreciated.
Thank you in advance