Miscellaneous
This is just a collection of random laws that I’ve found. Some of them are more useful than others.
Inclusion
$(a \wedge b) = a \quad = \quad (a \Rightarrow b)$
\[\begin{align} &(a \wedge b) = a && \\ &= ((a \wedge b) \wedge a) \vee ((\overline{a \wedge b}) \wedge \overline{a}) &&\text{Equality} \\ &= ((a \wedge b) \wedge a) \vee ((\overline{a} \vee \overline{b}) \wedge \overline{a}) &&\text{De Morgan's Law} \\ &= ((a \wedge a) \wedge b) \vee (\overline{a} \wedge (\overline{a} \vee \overline{b})) &&\text{Associativity and Commutativity} \\ &= (a \wedge b) \vee (\overline{a} \wedge (\overline{a} \vee \overline{b})) &&\text{Idempotent} \\ &= (a \wedge b) \vee \overline{a} &&\text{Absorption} \\ &= \overline{a} \vee (a \wedge b) &&\text{Commutativity} \\ &= \overline{a} \vee b &&\text{Simplification} \\ &= a \Rightarrow b &&\text{Material Implication} \end{align}\]$(a \vee b) = b \quad = \quad (a \Rightarrow b)$
\[\begin{align} &(a \vee b) = b && \\ &= ((a \vee b) \wedge b) \vee ((\overline{a \vee b}) \wedge \overline{b}) &&\text{Equality} \\ &= ((a \vee b) \wedge b) \vee ((\overline{a} \wedge \overline{b}) \wedge \overline{b}) &&\text{De Morgan's Law} \\ &= (b \wedge (b \vee a)) \vee ((\overline{b} \wedge \overline{b}) \wedge \overline{a}) &&\text{Associativity and Commutativity} \\ &= (b \wedge (b \vee a)) \vee (\overline{b} \wedge \overline{a}) &&\text{Idempotent} \\ &= b \vee (\overline{b} \wedge \overline{a}) &&\text{Absorption} \\ &= b \vee \overline{a} &&\text{Simplification} \\ &= \overline{a} \vee b &&\text{Commutativity} \\ &= a \Rightarrow b &&\text{Material Implication} \end{align}\]Discharge
$a \wedge (a \Rightarrow b) \quad = \quad (a \wedge b)$
\[\begin{align} &a \wedge (a \Rightarrow b) && \\ &= a \wedge (\overline{a} \vee b) &&\text{Material Implication} \\ &= a \wedge b &&\text{Simplification} \end{align}\]$a \Rightarrow (a \wedge b) \quad = \quad (a \Rightarrow b)$
\[\begin{align} &a \Rightarrow (a \wedge b) && \\ &= \overline{a} \vee (a \wedge b) &&\text{Material Implication} \\ &= \overline{a} \vee b &&\text{Simplification} \\ &= a \Rightarrow b &&\text{Material Implication} \end{align}\]Portation
$(a \wedge b) \Rightarrow c \quad = \quad a \Rightarrow (b \Rightarrow c)$
\[\begin{align} &(a \wedge b) \Rightarrow c && \\ &= (\overline{a \wedge b}) \vee c &&\text{Material Implication} \\ &= (\overline{a} \vee \overline{b}) \vee c &&\text{De Morgan's Law} \\ &= \overline{a} \vee (\overline{b} \vee c) &&\text{Associativity} \\ &= \overline{a} \vee (b \Rightarrow c) &&\text{Material Implication} \\ &= a \Rightarrow (b \Rightarrow c) &&\text{Material Implication} \end{align}\]Conflation
$(a \Rightarrow b) \wedge (c \Rightarrow d) \quad \Rightarrow \quad (a \wedge c) \Rightarrow (b \wedge d)$
\[\begin{align} &(a \Rightarrow b) \wedge (c \Rightarrow d) && \\ &= (\overline{a} \vee b) \wedge (\overline{c} \vee d) &&\text{Material Implication 2 times} \\ &\Rightarrow ((\overline{a} \vee b) \wedge (\overline{c} \vee d)) \vee (\overline{a} \vee \overline{c}) &&\text{Generalization} \\ &= ((\overline{a} \vee b) \vee (\overline{a} \vee \overline{c})) \wedge ((\overline{c} \vee d) \vee (\overline{a} \vee \overline{c})) &&\text{Distributivity} \\ &= ((\overline{a} \vee \overline{a}) \vee b \vee \overline{c}) \wedge (\overline{a} \vee (\overline{c} \vee \overline{c}) \vee d) &&\text{Associativity and Commutativity} \\ &= (\overline{a} \vee b \vee \overline{c}) \wedge (\overline{a} \vee \overline{c} \vee d) &&\text{Idempotent 2 times} \\ &= ((\overline{a} \vee \overline{c}) \vee b) \wedge ((\overline{a} \vee \overline{c}) \vee d) &&\text{Associativity and Commutativity} \\ &= (\overline{a} \vee \overline{c}) \vee (b \wedge d) &&\text{Distributivity} \\ &= (\overline{a \wedge c}) \vee (b \wedge d) &&\text{De Morgan's Law} \\ &= (a \wedge c) \Rightarrow (b \wedge d) &&\text{Material Implication} \end{align}\]$(a \Rightarrow b) \wedge (c \Rightarrow d) \quad \Rightarrow \quad (a \vee c) \Rightarrow (b \vee d)$
\[\begin{align} &(a \Rightarrow b) \wedge (c \Rightarrow d) && \\ &= (\overline{a} \vee b) \wedge (\overline{c} \vee d) &&\text{Material Implication 2 times} \\ &\Rightarrow ((\overline{a} \vee b) \wedge (\overline{c} \vee d)) \vee (b \vee d) &&\text{Generalization} \\ &= ((\overline{a} \vee b) \vee (b \vee d)) \wedge ((\overline{c} \vee d) \vee (b \vee d)) &&\text{Distributivity} \\ &= (\overline{a} \vee (b \vee b) \vee d) \wedge (b \vee \overline{c} \vee (d \vee d)) &&\text{Associativity and Commutativity} \\ &= (\overline{a} \vee b \vee d) \wedge (b \vee \overline{c} \vee d) &&\text{Idempotent 2 times} \\ &= (\overline{a} \vee (b \vee d)) \wedge (\overline{c} \vee (b \vee d)) &&\text{Associativity and Commutativity} \\ &= (\overline{a} \vee \overline{c}) \vee (b \wedge d) &&\text{Distributivity} \\ &= (\overline{a \wedge c}) \vee (b \wedge d) &&\text{De Morgan's Law} \\ &= (a \wedge c) \Rightarrow (b \wedge d) &&\text{Material Implication} \end{align}\]