# Given one of De Morgan's laws, prove the other from it using equivalences.

I have one of De Morgan's laws (in propositional logic). I would like to prove the other law from the first using a sequence of equivalences (Resolution).

One is not allowed to use truth tables or the particular De Morgans law which we are trying to prove (obviously).

How can this be done?

$$\lnot (A\land B)\equiv \lnot A \lor \lnot B$$

$$\lnot (A\lor B)\equiv \lnot A \land \lnot B$$

a set of resolution equivalences laws can be found on wikipedia

Let assume $$\lnot (A \land B) \equiv \lnot A \lor \lnot B$$.
Then from $$\lnot (A \lor B)$$ using Double Negation, we get $$\lnot (\lnot \lnot A \lor \lnot \lnot B) \equiv \lnot \lnot (\lnot A \land \lnot B)$$.