Proving De Morgan's Law with Natural Deduction 
Here is my attempt, but I'm really not sure if I've done it right;
as I'm just about getting the hang of Natural Deduction technique.  
Have I done it correctly? If not, where did I make errors and how should I do it?
Thank you in advance!
Sorry for the bad image quality; I'm bad at taking pictures.
 A: In the second part lines 2 and 7 are redundant, instead assume line 3.
Edit: As noticed by @GitGud, in the first part, line 6 must be $\neg\neg p \ (\neg I)$. So you need to add another line to get $p$ by $(\neg\neg E)$. Similar correction for line 10.
A: Just a little uncomfortable with step 6 of part 2, i.e., using three steps (1,4,5) to assert a contradiction - seems like too much intuition for my liking. If possible, would like to limit myself to just the { φ ∧ ¬φ } form of contradiction. Minimal intuition - an object cannot be both itself and not itself at the same time.
Is the following possible instead? {n}, where n is a natural number, denotes underlying dependencies.
{1}    1. ¬P ∨ ¬Q              premise
{2}    2. P ∧ Q                assumption for proof by contradiction
{2}    3. P                    2 ∧E
{2}    4. Q                    2 ∧E
{5}    5. ¬P                   assumption, 1st disjunct of ¬P ∨ ¬Q
{2,5}  6. P ∧ ¬P               3,5 ∧I
{7}    7. ¬Q                   assumption, 2nd disjunct of ¬P ∨ ¬Q
{2,7}  8. Q ∧ ¬Q               4,7 ∧I
{1,2}  9. (P ∧ ¬P) ∨ (Q ∧ ¬Q)  1,5,6,7,8 Discharge 5,7   i.e. ⊥ 
{1}    10. ¬(P ∧ Q)            2,9 proof by contradiction, discharge 2

Since step 10 is an outcome of the premise alone, therefore (¬P ∨ ¬Q) ⊢ ¬(P ∧ Q).
11 Feb:
Just an additional comment after some thought. Citing steps 1 (¬P ∨ ¬Q), 4(P) and 6(Q) to justify a contradiction is implicitly claiming that (¬P ∨ ¬Q) is in contradiction with (P ∧ Q) (i.e. conjunction of steps 4 and 6). But this contradiction is the very thing we're trying to prove. That's why I wasn't comfortable previously. Glad for comments/correction if any. Thanks!
A: The issue that @Wei brought up appears the most critical.  Here is a proof imitating the script as much as possible:

The blue boxes on lines 6 and 10 show that the justification would be indirect proof (IP) not negation introduction. @LoMaPh offers a different suggestion for fixing that. 
As LoMaPh points out one could skip lines 16 and 25, but it is not incorrect to leave them as they are.
The issue @Wei brought up is critical and is fixed in the red box. One needs to explicitly do a disjunction elimination on both disjuncts in line 15.
