Switch Between Bibliography and Appendix in fancyhdr

Question

What I am trying to accomplish is a fancyhdr that places Appendix or Bibliography in the header with one single definition. So if I am in the bibliography it should display Bibliography, but should display Appendix otherwise.

\documentclass{report}
\usepackage{fancyhdr}
\usepackage{etoolbox}

\newcommand{\backmatter}{
    \fancyhdr[C]{\ifstrequal{\leftmark}{\MakeUppercase{\bibname}}
        {\bibname}{\appendixname\ \thechapter}}

\begin{document}
   \backmatter       
   \bibliography{my.bib}
   \appendix
   \chapter{Appendix}
\end{document}

The ifstrequal always evaluates as false and so always produces Appendix in the header. Thank you in advance for your help.

EDIT: My new solution is as follows

\documentclass{report}
\usepackage{fancyhdr}
\usepackage{etoolbox}

\preptocmd{\bibliography}{\fancyhead[C]{\bibname}}{}{}
\preptocmd{\appendix}{\fancyhead[C]{\appendixname\ \thechapter}}{}{}

\begin{document}       
   \bibliography{my.bib}
   \appendix
   \chapter{Appendix}
\end{document}

However, when my bibliography hits the third page it displays Appendix again.

Edit 2: The following works

\documentclass{report}
\usepackage{fancyhdr}
\usepackage{etoolbox}

\preptocmd{\bibliography}{\cleardoublepage \fancyhead[C]{\bibname}}{}{}
\preptocmd{\appendix}{\cleardoublepage \fancyhead[C]{\appendixname\ \thechapter}}{}{}

\begin{document}       
   \bibliography{my.bib}
   \appendix
   \chapter{Appendix}
\end{document}

Answer 1

The main problem is that \leftmark doesn't contain \MakeUppercase{\bibname}, but a more complicated thing. There are also other expansion problems.

If your backmatter is in that form, I suggest

\newif\ifbiblio
\newcommand{\backmatter}{%
  \fancyhead[C]{\ifbiblio \bibname\else\appendixname\ \thechapter\fi}%
  \bibliotrue
}
\appto\appendix{\clearpage\bibliofalse}

The conditional will be true from \backmatter until after the page change for the start of the appendices.