\FigBox{%
\[\begin{array}{rcll}
(\alpha\land \beta) &\equiv& (\beta\land \alpha) \quad\mbox{commutativity of }\land\\
(\alpha\lor \beta) &\equiv& (\beta\lor \alpha) \quad\mbox{commutativity of }\lor\\
((\alpha\land \beta)\land \gamma) &\equiv& (\alpha\land (\beta\land \gamma))  \quad\mbox{associativity of }\land\\
((\alpha\lor \beta)\lor \gamma) &\equiv& (\alpha\lor (\beta\lor \gamma))  \quad\mbox{associativity of }\lor\\
\lnot(\lnot \alpha) &\equiv& \alpha \quad\mbox{double-negation elimination}\\
(\alpha\implies \beta) &\equiv& (\lnot \beta \implies \lnot \alpha) \quad\mbox{contraposition}\\
(\alpha\implies \beta) &\equiv& (\lnot \alpha \lor \beta) \quad\mbox{implication elimination}\\
(\alpha\lequiv \beta) &\equiv& ((\alpha\implies \beta)\land (\beta\implies \alpha)) \quad\mbox{biconditional elimination}\\
\lnot(\alpha\land \beta) &\equiv& (\lnot \alpha \lor \lnot \beta) \quad\mbox{De Morgan}\\
\lnot(\alpha\lor \beta) &\equiv& (\lnot \alpha \land \lnot \beta) \quad\mbox{De Morgan}\\
(\alpha\land (\beta\lor \gamma)) &\equiv& ((\alpha\land \beta)\lor (\alpha\land \gamma))\quad\mbox{distributivity of }\land\mbox{ over }\lor\\
(\alpha\lor (\beta\land \gamma)) &\equiv& ((\alpha\lor \beta)\land (\alpha\lor \gamma)) \quad\mbox{distributivity of }\lor\mbox{ over }\land
\end{array}
\]}



