Distributivity Continued
Now we prove some other useful distributivity laws.
$a \wedge (b \wedge c) \quad = \quad (a \wedge b) \wedge (a \wedge c)$
\[\begin{align} &a \wedge (b \wedge c) &&\\ &= (a \wedge a) \wedge (b \wedge c) &&\text{Idempotent} \\ &= (a \wedge b) \wedge (a \wedge c) &&\text{Commutativity and Associativity} \end{align}\]$a \vee (b \vee c) \quad = \quad (a \vee b) \vee (a \vee c)$
\[\begin{align} &a \vee (b \vee c) &&\\ &= (a \vee a) \vee (b \vee c) &&\text{Idempotent} \\ &= (a \vee b) \vee (a \vee c) &&\text{Commutativity and Associativity} \end{align}\]$a \Rightarrow (b \wedge c) \quad = \quad (a \Rightarrow b) \wedge (a \Rightarrow c)$
\[\begin{align} &a \Rightarrow (b \wedge c) && \\ &= \overline{a} \vee (b \wedge c) &&\text{Material Implication} \\ &= (\overline{a} \vee b) \wedge (\overline{a} \vee c) &&\text{Distributivity} \\ &= (a \Rightarrow b) \wedge (a \Rightarrow c) &&\text{Material Implication 2 times} \end{align}\]$a \Rightarrow (b \vee c) \quad = \quad (a \Rightarrow b) \vee (a \Rightarrow c)$
\[\begin{align} &a \Rightarrow (b \vee c) && \\ &= \overline{a} \vee (b \vee c) &&\text{Material Implication} \\ &= (\overline{a} \vee b) \vee (\overline{a} \vee c) &&\text{Distributivity} \\ &= (a \Rightarrow b) \vee (a \Rightarrow c) &&\text{Material Implication 2 times} \end{align}\]$(a \wedge b) \Rightarrow c \quad = \quad (a \Rightarrow c) \vee (b \Rightarrow c)$
\[\begin{align} &(a \wedge b) \Rightarrow c && \\ &= (\overline{a \wedge b}) \vee c &&\text{Material Implication} \\ &= (\overline{a} \vee \overline{b}) \vee c &&\text{De Morgan's Law} \\ &= (\overline{a} \vee c) \vee (\overline{b} \vee c) &&\text{Distributivity} \\ &= (a \Rightarrow c) \vee (b \Rightarrow c) &&\text{Material Implication 2 times} \end{align}\]$(a \vee b) \Rightarrow c \quad = \quad (a \Rightarrow c) \wedge (b \Rightarrow c)$
\[\begin{align} &(a \vee b) \Rightarrow c && \\ &= (\overline{a \vee b}) \vee c &&\text{Material Implication} \\ &= (\overline{a} \wedge \overline{b}) \vee c &&\text{De Morgan's Law} \\ &= (\overline{a} \vee c) \wedge (\overline{b} \vee c) &&\text{Distributivity} \\ &= (a \Rightarrow c) \wedge (b \Rightarrow c) &&\text{Material Implication 2 times} \end{align}\]$a \vee (b \Rightarrow c) \quad = \quad (a \vee b) \Rightarrow (a \vee c)$
\[\begin{align} &a \vee (b \Rightarrow c) &&\\ &= a \vee (\overline{b} \vee c) &&\text{Material Implication} \\ &= (a \vee \overline{b}) \vee c &&\text{Associativity} \\ &= (a \vee (\overline{a} \wedge \overline{b})) \vee c &&\text{Simplification} \\ &= (\overline{a} \wedge \overline{b}) \vee (a \vee c) &&\text{Commutativity and Associativity} \\ &= (\overline{a \vee b}) \vee (a \vee c) &&\text{De Morgan's Law} \\ &= (a \vee b) \Rightarrow (a \vee c) &&\text{Material Implication} \end{align}\]$a \vee (b = c) \quad = \quad (a \vee b) = (a \vee c)$
\[\begin{align} &a \vee (b = c) &&\\ &= a \vee ((b \Rightarrow c) \wedge (b \impliedby c)) &&\text{Double Implication} \\ &= a \vee ((b \Rightarrow c) \wedge (c \Rightarrow b)) &&\text{Mirror} \\ &= (a \vee (b \Rightarrow c)) \wedge (a \vee (c \Rightarrow b)) &&\text{Distributivity of } \ \vee \\ &= ((a \vee b) \Rightarrow (a \vee c)) \wedge ((a \vee c) \Rightarrow (a \vee b)) &&\text{Distributivity of} \ \vee \ \text{into} \ \Rightarrow \\ &= ((a \vee b) \Rightarrow (a \vee c)) \wedge ((a \vee b) \impliedby (a \vee c)) &&\text{Mirror} \\ &= ((a \vee b) = (a \vee c)) &&\text{Double Implication} \end{align}\]$a \Rightarrow (b \Rightarrow c) \quad = \quad (a \Rightarrow b) \Rightarrow (a \Rightarrow c)$
\[\begin{align} &a \Rightarrow (b \Rightarrow c) &&\\ &= \overline{a} \vee (b \Rightarrow c) &&\text{Material Implication} \\ &= (\overline{a} \vee b) \Rightarrow (\overline{a} \vee c) &&\text{Distributivity of} \ \vee \text{into} \ \Rightarrow \\ &= (a \Rightarrow b) \Rightarrow (a \Rightarrow c) &&\text{Material Implication 2 times} \end{align}\]$a \Rightarrow (b = c) \quad = \quad (a \Rightarrow b) = (a \Rightarrow c)$
\[\begin{align} &a \Rightarrow (b = c) &&\\ &= \overline{a} \vee (b = c) &&\text{Material Implication} \\ &= (\overline{a} \vee b) = (\overline{a} \vee c) &&\text{Distributivity of} \ \vee \text{into} \ = \\ &= (a \Rightarrow b) = (a \Rightarrow c) &&\text{Material Implication 2 times} \end{align}\]