We start by computing $\skel$:
  \begin{compactItemize}
  	\item $e_{0}\Leftrightarrow(g(a)=c)$
  	\item $e_{1}\Leftrightarrow(f(g(a))=f(c))$
  	\item $e_{2}\Leftrightarrow(g(a)=d)$
  	\item $e_{3}\Leftrightarrow(c=d)$
  \end{compactItemize}
  $$ \skel = e_{0} \land \neg e_{1} \land e_{2} \land \neg e_{3} $$


  \hspace{-0.09cm}\scalebox{0.85}{
  	\begin{dplltabular}{4}
  		\dpllStep{1|2|3|4}
  		\dpllDecL{0|0|0|0}
  		\dpllAssi{ - |$e_{0}$|$e_{0}, \lnot e_{3}$|$e_{0}, \lnot e_{3}, \lnot e_{1}$}
  		\dpllClause{1}{$e_{0}$}{$e_{0}$|\done|\done|\done}
  		\dpllClause{2}{$\lnot e_{1}, e_{2}$}{$\lnot e_{1}, e_{2}$|$\lnot e_{1}, e_{2}$|$\lnot e_{1}, e_{2}$|\done}
  		\dpllClause{3}{$\lnot e_{3}$}{$\lnot e_{3}$|$\lnot e_{3}$|\done|\done}
  		\dpllBCP{$e_{0}$|$\lnot e_{3}$| - | - }
  		\dpllPL{ - | - |$\lnot e_{1}$| - }
  		\dpllDeci{ - | - | - | - }
  	\end{dplltabular}
  }

  A satisfying assignment $e_0 \land \neg e_1 \land \neg e_3$ has been found and
  we have to check consistency of $(g(a) = c) \land (f(g(a)) \neq f(c)) \land (c \neq d) $:

  \begin{align*}
  	&\{g(a), c\}, \{f(g(a))\}, \{f(c)\}, \{d\}\\
  	&\{g(a), c\}, \{d\}, \{f(c), f(g(a))\}
  \end{align*}

  Congruence Closure returns UNSAT because of: $(f(g(a)) \neq f(c)) $.
  We therefore add $\neg e_0 \lor e_1 \lor e_3$ as a blocking clause and continue.
  \hspace{-0.09cm}\scalebox{0.85}{
  	\begin{dplltabular}{5}
  		\dpllStep{5|6|7|8|9}
  		\dpllDecL{0|0|0|0|0}
  		\dpllAssi{ - |$e_{0}$|$e_{0}, \lnot e_{3}$|$e_{0}, \lnot e_{3}, e_{1}$|\makecell{$e_{0}, \lnot e_{3}, e_{1}, $ \\ $e_{2}$}}
  		\dpllClause{1}{$e_{0}$}{$e_{0}$|\done|\done|\done|\done}
  		\dpllClause{2}{$\lnot e_{1}, e_{2}$}{$\lnot e_{1}, e_{2}$|$\lnot e_{1}, e_{2}$|$\lnot e_{1}, e_{2}$|$e_{2}$|\done}
  		\dpllClause{3}{$\lnot e_{3}$}{$\lnot e_{3}$|$\lnot e_{3}$|\done|\done|\done}
  		\dpllClause{4}{$\lnot e_{0}, e_{3}, e_{1}$}{$\lnot e_{0}, e_{3}, e_{1}$|$e_{3}, e_{1}$|$e_{1}$|\done|\done}
  		\dpllBCP{$e_{0}$|$\lnot e_{3}$|$e_{1}$|$e_{2}$| - }
  		\dpllPL{ - | - | - | - | - }
  		\dpllDeci{ - | - | - | - | - }
  	\end{dplltabular}
  }

  We have to check consistency for $(g(a) = c) \land (f(g(a)) = f(c)) \land (g(a) = d) \land (c \neq d) $:

  \begin{align*}
  	&\{g(a), c\}, \{f(g(a)), f(c)\}, \{g(a), d\}\\
  	&\{f(g(a)), f(c)\}, \{c, d, g(a)\}
  \end{align*}

  Congruence Closure returns UNSAT  because of: $(c \neq d) $\\
  We therefore add $\neg e_0 \lor e_3 \lor \neg e_1 \lor \neg e_2$ as a blocking clause and continue.
  \hspace{-0.09cm}\scalebox{0.80}{
  	\begin{dplltabular}{5}
  		\dpllStep{10|11|12|13|14}
  		\dpllDecL{0|0|0|0|0}
  		\dpllAssi{ - |$e_{0}$|$e_{0}, \lnot e_{3}$|$e_{0}, \lnot e_{3}, e_{1}$|\makecell{$e_{0}, \lnot e_{3}, e_{1}, $ \\ $\lnot e_{2}$}}
  		\dpllClause{1}{$e_{0}$}{$e_{0}$|\done|\done|\done|\done}
  		\dpllClause{2}{$\lnot e_{1}, e_{2}$}{$\lnot e_{1}, e_{2}$|$\lnot e_{1}, e_{2}$|$\lnot e_{1}, e_{2}$|$e_{2}$|\conflict}
  		\dpllClause{3}{$\lnot e_{3}$}{$\lnot e_{3}$|$\lnot e_{3}$|\done|\done|\done}
  		\dpllClause{4}{$\lnot e_{0}, e_{3}, e_{1}$}{$\lnot e_{0}, e_{3}, e_{1}$|$e_{3}, e_{1}$|$e_{1}$|\done|\done}
  		\dpllClause{5}{$\lnot e_{0}, e_{3}, \lnot e_{1}, \lnot e_{2}$}{\makecell{$\lnot e_{0}, e_{3}, \lnot e_{1}, $ \\ $\lnot e_{2}$}|$e_{3}, \lnot e_{1}, \lnot e_{2}$|$\lnot e_{1}, \lnot e_{2}$|$\lnot e_{2}$|\done}
  		\dpllBCP{$e_{0}$|$\lnot e_{3}$|$e_{1}$|$\lnot e_{2}$| - }
  		\dpllPL{ - | - | - | - | - }
  		\dpllDeci{ - | - | - | - |UNSAT}
  	\end{dplltabular}
  }

  Conflict in step 14\\
  \scalebox{0.75}{

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  }
  \begin{prooftree}
  	\AxiomC{$2. \; \lnot e_{1} \lor e_{2}$}
  	\AxiomC{$5. \; \lnot e_{0} \lor e_{3} \lor \lnot e_{1} \lor \lnot e_{2}$}
  	\BinaryInfC{$\lnot e_{1} \lor \lnot e_{0} \lor e_{3}$}
  	\AxiomC{$4. \; \lnot e_{0} \lor e_{3} \lor e_{1}$}
  	\BinaryInfC{$\lnot e_{0} \lor e_{3}$}
  	\AxiomC{$1. \; e_{0}$}
  	\BinaryInfC{$e_{3}$}
  	\AxiomC{$3. \; \lnot e_{3}$}
  	\BinaryInfC{$\bot$}
  \end{prooftree}

  Since the SAT solver cannot find a satisfying assignment we are done and conclude that $\varphi$ is UNSAT.