rule of replacement

!Rules of inference

!Tautology

!Name

(p \vee q) \vee r\\

\therefore \overline{p \vee (q \vee r)} \\

\end{align}

|$((p\; \backslash vee\; q)\; \backslash vee\; r)\; \backslash rightarrow\; (p\; \backslash vee\; (q\; \backslash vee\; r))$

|Associative

p \wedge q\\

\therefore \overline{q \wedge p} \\

\end{align}

|$(p\; \backslash wedge\; q)\; \backslash rightarrow\; (q\; \backslash wedge\; p)$

|Commutative

(p \wedge q) \rightarrow r\\

\therefore \overline{p \rightarrow (q \rightarrow r)} \\

\end{align}

|$((p\; \backslash wedge\; q)\; \backslash rightarrow\; r)\; \backslash rightarrow\; (p\; \backslash rightarrow\; (q\; \backslash rightarrow\; r))$

|Exportation

p \rightarrow q\\

\therefore \overline{\neg q \rightarrow \neg p} \\

\end{align}

|$(p\; \backslash rightarrow\; q)\; \backslash rightarrow\; (\backslash neg\; q\; \backslash rightarrow\; \backslash neg\; p)$

|Transposition or contraposition law

p \rightarrow q\\

\therefore \overline{\neg p \vee q} \\

\end{align}

|$(p\; \backslash rightarrow\; q)\; \backslash rightarrow\; (\backslash neg\; p\; \backslash vee\; q)$

|Material implication

(p \vee q) \wedge r\\

\therefore \overline{(p \wedge r) \vee (q \wedge r)} \\

\end{align}

|$((p\; \backslash vee\; q)\; \backslash wedge\; r)\; \backslash rightarrow\; ((p\; \backslash wedge\; r)\; \backslash vee\; (q\; \backslash wedge\; r))$

|Distributive

p\\

q\\

\therefore \overline{p \wedge q} \\

\end{align}

|$((p)\; \backslash wedge\; (q))\; \backslash rightarrow\; (p\; \backslash wedge\; q)$

|Conjunction

p\\

\therefore \overline{\neg \neg p} \\

\end{align}

|$p\; \backslash rightarrow\; (\backslash neg\; \backslash neg\; p)$

|Double negation introduction

{\neg \neg p}\\

\therefore \overline p\\

\end{align}

|$(\backslash neg\; \backslash neg\; p)\; \backslash rightarrow\; p$

|Double negation elimination