De Morgan's Laws in 39 steps (instead of truth tables)

CURRENT LINE NUMBER	DEPENDS ON	CURRENT LINE	DERIVED BY
1.	1	¬ (P ∨ Q)	Assumption [discharged line 11]
2.	2	P	Assumption [discharged line 5]
3.	2	P ∨ Q	2, ∨I
4.	1, 2	(P ∨ Q) & ¬ (P ∨ Q)	1, 3 &I
5.	1	¬ P	2, 4 ¬I
6.	6	Q	Assumption [discharged line 9]
7.	6	P ∨ Q	6, ∨I
8.	1, 6	(P ∨ Q) & ¬ (P ∨ Q)	1, 7, &I
9.	1	¬ Q	6, 8 ¬I
10.	1	¬ P & ¬ Q	5, 9 &I
11.		¬ (P ∨ Q) ⊃ ¬ P & ¬ Q	1, 10 ⊃I
12.	12	¬ P & ¬ Q	Assumption [discharged line 24]
13.	12	¬ P	12 &E
14.	14	P ∨ Q	Assumption [discharged line 24]
15.	15	P	Assumption [discharged line 21]
16.	16	¬ Q	Assumption [discharged line 18]
17.	12, 15	P & ¬ P	13, 15 &I
18.	12, 15	¬ ¬ Q	16,17 ¬I
19.	12, 15	Q	18 ¬E
20.	20	Q	Assumption [discharged line 21]
21.	12, 14	Q	14, 15, 19, 20, 20 ∨E
22.	12	¬ Q	12, &E
23.	12, 14	Q & ¬ Q	21, 22 &I
24.	12	¬ (P ∨ Q)	14, 23 ¬E
25.		¬ P & ¬ Q ⊃ ¬ (P ∨ Q)	12, 24 ⊃I
26.		(¬ (P ∨ Q) ⊃ ¬ P & ¬ Q)
		& (¬ P & ¬ Q ⊃ ¬ (P ∨ Q))	11, 25 &I
27.		¬ P & ¬ Q ≡ ¬ (P ∨ Q)	26 ≡I
28.	28	¬ (P & Q)	Assumption [discharged line 32]
29.	28	¬ (¬ ¬ P & ¬ ¬ Q)	28 DN
30.	28	¬ (¬ (¬ P ∨ ¬ Q))	29, 25 ⊃E
31.	28	¬ P ∨ ¬ Q	30, ¬E
32.		¬ (P & Q) ⊃ ¬ P ∨ ¬ Q	28, 31 ⊃I
33.	33	¬ P ∨ ¬ Q	Assumption [discharged line 37]
34.	33	¬ ¬ (¬ P ∨ ¬ Q)	33 DN
35.	33	¬ (¬ ¬ P & ¬ ¬ Q)	34, 11 ⊃E
36.	33	¬ ( P & Q)	35 ¬E
37.		¬ P ∨ ¬ Q ⊃ ¬ ( P & Q)	33, 36 ⊃I
38.		(¬ (P & Q) ⊃ ¬ P ∨ ¬ Q)
		& (¬ P ∨ ¬ Q ⊃ ¬ ( P & Q))	32, 37 &I
39.		¬ P ∨ ¬ Q ≡ ¬ ( P & Q)	38 ≡I