Functionally Complete Logical Connectives/Negation, Conjunction, Disjunction and Implication

Theorem
The set of logical connectives:
 * $\left\{{\neg, \land, \lor, \implies}\right\}$: Not, And, Or and Implies

is functionally complete.

Proof
From Functional Completeness over Finite Number of Arguments, it suffices to consider binary truth functions.

From Count of Truth Functions, there are $16$ of these.

These are enumerated in Binary Truth Functions, and are analysed in turn as follows.

Constant Functions
There are two constant functions:

From Biconditional Properties and Exclusive Or Properties:

where $\oplus$ and $\iff$ denote exclusive or and biconditional respectively.

So both of the constant functions can be expressed in terms of $\oplus$ and $\iff$.

Equivalence and Non-Equivalence
By definition of exclusive or:
 * $p \oplus q \dashv \vdash \neg \left({p \iff q}\right)$.

Thus $\oplus$ can be expressed in terms of $\neg$ and $\iff$.

By definition of biconditional:
 * $p \iff q \dashv \vdash \left({p \implies q}\right) \land \left({q \implies p}\right)$

Thus $\iff$ can be expressed in terms of $\implies$ and $\land$.

Projections and Negated Projections
There are two projections:
 * $\operatorname{pr}_1 \left({p, q}\right) = p$
 * $\operatorname{pr}_2 \left({p, q}\right) = q$

We note that:
 * $p \land p \dashv \vdash p \dashv \vdash \operatorname{pr}_1 \left({p, q}\right)$
 * $p \lor p \dashv \vdash p \dashv \vdash \operatorname{pr}_1 \left({p, q}\right)$

and similarly for $\operatorname{pr}_2 \left({p, q}\right)$.

So the projections can be expressed in terms of either $\land$ or $\lor$.

There are two negated projections:
 * $\overline {\operatorname{pr}_1} \left({p, q}\right) = \neg p$
 * $\overline {\operatorname{pr}_2} \left({p, q}\right) = \neg q$

It immediately follows from the above that these can be expressed in terms of either:
 * $\neg$ and $\land$

or:
 * $\neg$ and $\lor$

NAND and NOR
There are the NAND and NOR operators:
 * $p \uparrow q$
 * $p \downarrow q$

By definition of NAND:
 * $p \uparrow q \dashv \vdash \neg \left({p \land q}\right)$

By definition of NOR:
 * $p \downarrow q \dashv \vdash \neg \left({p \lor q}\right)$

So:
 * $\uparrow$ can be expressed in terms of $\neg$ and $\land$
 * $\downarrow$ can be expressed in terms of $\neg$ and $\lor$

Conjunction, Disjunction, Conditional
There are the conjunction and disjunction operators:
 * $p \land q$
 * $p \lor q$

There are two conditionals:
 * $p \implies q$
 * $q \implies p$

There are two negated conditionals:
 * $\neg \left({p \implies q}\right)$
 * $\neg \left({q \implies p}\right)$

All of the above are already expressed in terms of $\neg, \land, \lor, \implies$.

Thus all sixteen of the Binary Truth Functions can be expressed in terms of:
 * $\neg, \land, \lor, \implies$

That is:
 * $\left\{{\neg, \land, \lor, \implies}\right\}$

is functionally complete.