I have this argument I'm trying to prove:
¬(A ∨ B)
_
¬A ∧ ¬B
We have two expressions which are equivalent to eachother. I know they are because they have the same truth values if you put both into a truth table. For formal proofs the rules I know are:
∧ Intro
∧ Elim
∨ Intro
∨ Elim
¬ Intro
¬ Elim
⊥ Intro
⊥ Elim
So I have to use those rules to prove this argument. From what I can see, there's nothing I can do with the premise because the main connective is the negation and I can't do anything with a single negation therefore the only rule I can start out with is ∧ Intro. The only problem is that I have no idea what to do next or if I'm even using the correct rule.
The thing is, I can formally prove this if it was the other way around:
¬A ∧ ¬B
_
¬(A ∨ B)
I just use ¬Intro on the bottom and then ∨ Elim and ∧ Elim proving that both A and B will lead to a contradiction. I'm just having problems proving it the other way around.
You start by assuming $A$, then introduce the $\lor$ obtaining $A\lor B$ so, using the hypothesis $\lnot(A\lor B)$, you get $\bot$, so you can discard the assumption $A$ and derive $\lnot A$. Do the same for $B$ and get $\lnot B$. So you got $\lnot A\land\lnot B$