De Morgan's Laws imply Uniquely Complemented Lattice is Boolean Lattice

Theorem
Let $\left({S,\wedge,\vee,\preceq}\right)$ be a Definition:Uniquely Complemented Lattice.

Then the following are equivalent:

$(1)\quad \forall p, q \in S: \neg p \vee \neg q = \neg \left({p \wedge q}\right)$

$(2)\quad \forall p, q \in S: \neg p \wedge \neg q = \neg \left({p \vee q}\right)$

$(3)\quad \forall p, q \in S: p \preceq q \iff \neg q \preceq \neg p$

$(4)\quad \left({S,\wedge,\vee,\preceq}\right)$ is a Definition:Distributive Lattice.

$(1)$ implies $(2)$
Suppose:
 * $\forall p, q \in S: \neg p \vee \neg q = \neg \left({p \wedge q}\right)$

Let $x, y \in S$.

Then $\neg \neg x \vee \neg \neg y = \neg \left({\neg x \wedge \neg y}\right)$.

By Complement of Complement in Uniquely Complemented Lattice, $\neg \neg x = x$ and $\neg \neg y = y$.

Thus:
 * $x \vee y = \neg \left({\neg x \wedge \neg y}\right)$.

Taking complements of both sides:
 * $\neg \left({x \vee y}\right) = \neg \neg \left({\neg x \wedge \neg y}\right)$

Again applying Complement of Complement in Uniquely Complemented Lattice:


 * $\neg \left({x \vee y}\right) = \neg x \wedge \neg y$

$(2)$ implies $(1)$
By Dual Pairs (Order Theory), $\wedge$ and $\vee$ are dual.

Thus this implication follows from the above by Duality.

$(1)$ implies $(3)$
By the definition of a lattice:
 * $p \preceq q \iff p \vee q = q$

Applying this to $\neg q$ and $\neg p$:
 * $\neg q \preceq \neg p \iff \neg q \vee \neg p = \neg p$

By $(1)$:
 * $\neg q \vee \neg p = \neg \left({q \wedge p}\right)$

So:
 * $\neg q \preceq \neg p \iff \neg \left({q \wedge p}\right) = \neg p$

Taking the complements of both sides of the equation on the right, and applying Complement of Complement in Uniquely Complemented Lattice:
 * $\neg q \preceq \neg p \iff \left({q \wedge p}\right) = p$

But the right side is equivalent to $p \preceq q$

Therefore:
 * $\neg q \preceq \neg p \iff p \preceq q$

$(3)$ implies $(1)$
Suppose that $p \preceq q \iff \neg q \preceq \neg p$

By the definition of join:
 * $\neg p, \neg q \preceq \neg p \vee \neg q$

Thus $\neg \left({\neg p \vee \neg q}\right) \preceq p, q$.

By the definition of meet:
 * $\neg \left({\neg p \vee \neg q}\right) \preceq p \wedge q$

Thus:
 * $\neg\left({p \wedge q}\right) \preceq \neg\neg \left({\neg p \vee \neg q}\right)$

By Complement of Complement in Uniquely Complemented Lattice: $*\quad \neg\left({p \wedge q}\right) \preceq \neg p \vee \neg q$

Dually:
 * $\neg x \wedge \neg y \preceq \neg \left({x \vee y}\right)$

Letting $x = \neg p$ and $y = \neg q$:
 * $\neg \neg p \wedge \neg \neg q \preceq \neg \left({\neg p \vee \neg q}\right)$

By Complement of Complement in Uniquely Complemented Lattice:
 * $p \wedge q \preceq \neg \left({\neg p \vee \neg q}\right)$

By the premise and Complement of Complement in Uniquely Complemented Lattice, then: $**\quad \neg p \vee \neg q \preceq \neg \left({p \wedge q}\right)$

By $*$ and $**$: $\quad \neg\left({p \wedge q}\right) = \neg p \vee \neg q$

$(1)$, $(2)$, and $(3)$ together imply $(4)$
$b, c \preceq b \vee c$, so
 * $a \wedge b \preceq a \wedge (b \vee c)$
 * $a \wedge c \preceq a \wedge (b \vee c)$

Thus:
 * $\neg ( a \wedge (b \vee c) ) \preceq \neg (a \wedge b)$
 * $\neg ( a \wedge (b \vee c) ) \preceq \neg (a \wedge c)$

Thus:
 * $\neg (a \wedge (b \vee c)) \preceq (\neg (a \wedge b)) \wedge (\neg (a \wedge c))$

So:
 * $\neg (a \wedge (b \vee c)) \preceq \neg ((a \wedge b) \vee (a \wedge c))$

So:
 * $(a \wedge b) \vee (a \wedge c) \preceq a \wedge (b \vee c)$