I have been given a list of statements which are provably equivalent, and it is my job to prove them. They are all quite similar in structure, but I'm having a difficult time proving even the first one, given the rules provided. I'm confident I can do the rest, if I can see the mechanics of the first in action.
The statement is:
$$ \neg (p \wedge q) \leftrightarrow \neg q \vee \neg p $$
The only rules I have at my disposal are:
$$ \wedge (AND, \wedge i - introduction, \wedge e - elimination) $$ $$ \vee (OR, \vee i - introduction, \vee i - elimination) $$ $$ \rightarrow (implication, \rightarrow i - introduction, \rightarrow e - elimination) $$ $$ \neg (negation, \neg i - introduction, \neg e - elimination) $$ $$ \bot (contradiction) $$ $$ \neg\neg (double negation) $$
Now, I know from other logic courses and books that De Morgans laws state that the negation of a conjunction is the disjunction of the negations, but how to I prove this step by step in a proof given the rules above. I keep running into the need to have a rule that distributes negation.