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\documentclass[10pt,a4paper]{article}
\usepackage[utf8]{inputenc}

\renewcommand{\labelenumii}{\theenumii}
\renewcommand{\theenumii}{\theenumi.\arabic{enumii}.}

\begin{document}
\begin{enumerate}
    \item $p \vdash p \lor q $ explanation  
    \begin{enumerate}
        \item $p \Rightarrow q$ explanation
    \end{enumerate}
\item $p \Rightarrow p \lor q$ explanation
\end{enumerate}
\end{document}

I'm trying to align the three explanations to the right as if I'm using a tabular environment but chose not to use tabular because I couldn't figure out how to get enumerate to work with it and I couldn't get the nice indentation on line 1.1 that you see in the proof

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enter image description here

\documentclass[10pt,a4paper]{article}
\usepackage[utf8]{inputenc}

\renewcommand{\labelenumii}{\theenumii}
\renewcommand{\theenumii}{\theenumi.\arabic{enumii}.}

\newcommand\explanation[1]{%
  \ifhmode\unskip\fi\hspace*{1em plus 1fill}%
\parbox[t]{6cm}{#1}}

\begin{document}
\begin{enumerate}
    \item $p \vdash p \lor q $ \explanation{
 zz z z z z zzz  blah balah  
 zz z z z z zzz  blah balah  
 zz z z z z zzz  blah balah} 
    \begin{enumerate}
        \item $p \Rightarrow q$ \explanation{zzzz}
    \end{enumerate}
\item $p \Rightarrow p \lor q$ \explanation{yet more comments
yet more comments yet more comments yet more comments}
\end{enumerate}
\end{document}
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