Redefine label in enumerate list

Question

First: Some days ago I posted this question on The LaTeX Community but I have not gotten an answer so I post it here.

Consider the following MWE:

\documentclass[danish]{article}
\usepackage{babel}
\usepackage{mathtools}
\usepackage{enumitem}
\usepackage{hyperref}

\mathtoolsset{showonlyrefs}
\newcommand*{\hv}[1]{\label{eqn:#1}\stackrel{\mathclap{(#1)}}{=}}
\newcommand*{\fk}[1]{\item[\eqref{eqn:#1}]}

\begin{document}

We have
\begin{align}
  \Delta y
  &\hv{1} m(x_{0}+h) - m(x_{0})\\
  &\hv{2} \frac{f(x_{0}+h)}{g(x_{0}+h)} - \frac{f(x_{0})}{g(x_{0})}\\
  &\hv{3} \frac{f(x_{0}+h) \cdot g(x_{0})}{g(x_{0}+h) \cdot g(x_{0})} - \frac{f(x_{0}) \cdot g(x_{0}+h)}{g(x_{0}) \cdot g(x_{0}+h)}\\
  &= \dots
\end{align}
Explanations:
\begin{enumerate}
  \fk{1} A
  \fk{2} B
  \fk{3} C
\end{enumerate}

\end{document}

Issues:

1. This way of refering to the lines in the align expression seems wierd. Is there a better way of doing this?
2. I will like the labels (1), (2), ... to be printed above = only and not flushed right too. I there a way of doing this?

Ad 2. I have tried putting \notag at the end of each line but then I get the following error:

Package amsmath Error: Multiple \label's: label 'eqn:1' will be lost.

Answer 1

I initially thought of suppressing the equation numbering, and using a custom numbering, but it turns out that \refstepcounter{...} was being ignored at runtime inside the math environment.

Managed to get a workaround, which produces the following output. I started the equation numbering at 10, to make sure the numbers correlated.

The key was in the label being assigned inside \makeatletter to \makeatother via the \ltx@label variable. The base numbering was suppressed via \begin{align*} to \end{align*} math environment.

The following is a sample output:

Which was produced with the following code:

\documentclass[danish]{article}
\usepackage{babel}
\usepackage{mathtools}
\usepackage{enumitem}
\usepackage{hyperref}

%\usepackage[color]{showkeys}
\mathtoolsset{showonlyrefs}

\setcounter{equation}{10}

\makeatletter
\newcommand*{\hv}[1]{%
    \refstepcounter{equation}%
    \ltx@label{EQ:#1}% Label%
    \stackrel{\mathclap{(\theequation)}}{=}%
}
\makeatother
\newcommand*{\fk}[1]{\item[\eqref{EQ:#1}]}

\begin{document}
We have
\begin{align*}
  \Delta y  &\hv{A} m(x_{0}+h) - m(x_{0})\\
            &\hv{B} \frac{f(x_{0}+h)}{g(x_{0}+h)} - \frac{f(x_{0})}{g(x_{0})}\\
            &\hv{C} \frac{f(x_{0}+h) \cdot g(x_{0})}{g(x_{0}+h) \cdot g(x_{0})} - \frac{f(x_{0}) \cdot g(x_{0}+h)}{g(x_{0}) \cdot g(x_{0}+h)}\\
            &= \dots
\end{align*}
Explanations:
\begin{enumerate}
  \fk{A} DESCA
  \fk{B} DESCB 
  \fk{C} DESCC
\end{enumerate}
\end{document}