documentclass{amsart}

usepackage{hyperref}

newtheorem{theorem}{Theorem}

begin{document}

begin{theorem}
label{bigthm}
begin{itemize}
item A claim.
item Another claim.
end{itemize}
end{theorem}

Let's prove Theorem~ref{bigthm}.

end{document}

pdfTeX warning (dest): name{theorem.1} has been
referenced but does not exist, replaced by a fixed one

begin{theorem}
label{bigthm}leavevmode
begin{itemize}[...]

begin{theorem}
phantomsectionlabel{bigthm}
begin{itemize}[...]

Let's prove~autoref{bigthm}.

documentclass{amsart}



usepackage{hyperref}
newtheorem{theorem}{Theorem}

makeatletter
newcommand{fakephantomsection}{%
  Hy@MakeCurrentHref{@currenvir.theHy@linkcounter}
  Hy@raisedlink{hyper@anchorstart{@currentHref}hyper@anchorend}%
}
makeatother



begin{document}

tableofcontents

clearpage
section{Foo}
begin{theorem}
fakephantomsectionlabel{bigthm}
begin{itemize}
item A claim.
item Another claim.
end{itemize}
end{theorem}

Let's prove Theorem~ref{bigthm} autoref{bigthm}.

end{document}

newcommand*{fakephantomsection}{%
  Hy@GlobalStepCountHy@linkcounter%
  Hy@MakeCurrentHref{@currenvir.theHy@linkcounter}%
  Hy@raisedlink{hyper@anchorstart{@currentHref}hyper@anchorend}%
}