1. ~(A v B) v ~(C v D)
2. (E v F) -> D   /     A -> ~E
-------------------------------
3.  (~A & ~B) v (~C & ~D)                       1    De Morgan's Law
4.  [(~A & ~B) v ~C] & [(~A & ~B) v ~D]         3    Distribution
5.  [(~A & ~B) v ~D] & [(~A & ~B) v ~C]         4    Commutation
6.  (~A & ~B) v ~D                              5    Simplification
7.  ~D v (~A & ~B)                              6    Commutation
8.  (~D v ~A) & (~D v ~B)                       7    Distribution
9.  ~D v ~A                                     8    Simplification
10. ~A v ~D                                     9    Commutation
11. A -> ~D                                    10    Material Implication
12. ~(E v F) v D                                2    Material Implication
13. (~E & ~F) v D                              12    De Morgan's Law
14. D v (~E & ~F)                              13    Commutation
15. (D v ~E) & (D v ~F)                        14    Distribution
16. D v ~E                                     15    Simplification
17. ~~D v ~E                                   16    Double Negation
18. ~D -> ~E                                   17    Material Implication
19. A -> ~E                                 11,18    Hypothetical Syllogism

1. ~(p <--> s)
2. (p v q) -> r
3. (t v ~s) -> (~s & ~r) Therefore: ~t & ~p
-------------------------------------------------
4.   ~[(p & s) v (~p & ~s)]                   1    Material Equivalence
5.   ~(p & s) & ~(~p & ~s)                    4    De Morgan's Law
6.   ~(~p & ~s) & ~(p & s)                    5    Commutation
7.   ~(p & s)                                 5    Simplification
8.   ~p v ~s                                  7    De Morgan's Law     
9.   ~s v ~p                                  8    Commutation
10.  s -> ~p                                  9    Material Implication
11.  ~(~p & ~s)                               6    Simplification
12.  ~~p v ~~s                                11   De Morgan's Law
13.  ~~p v s                                  12   Double Negation
14.  ~p -> s                                  13   Material Implication
------
15.  ~(p v q) v r                             2    Material Implication
16.  (~p & ~q) v r                            15   De Morgan's Law
17.  r v (~p & ~q)                            16   Commutation
18.  (r v ~p) & (r v ~q)                      17   Distribution
19.  r v ~p                                   18   Simplification
20.  ~p v r                                   19   Commutation
21.  p -> r                                   20   Material Implication
------
22.  ~(t v ~s) v (~s & ~r)                    3    Material Implication
23.  (~t & ~~s) v (~s & ~r)                   22   De Morgan's Law
24.  (~t & s) v (~s & ~r)                     23   Double Negation
25.  [(~t & s) v ~s] & [(~t & s) v ~r]        24   Distribution
26.  [(~t & s) v ~r] & [(~t & s) v ~s]        25   Commutation
27.  (~t & s) v ~s                            25   Simplification
28.  ~s v (~t & s)                            27   Commutation
29.  s -> (~t & s)                            28   Material Implication
30. (~t & s) v ~r                             26   Simplification   
31. ~r v (~t & s)                             30   Commutation
32. r -> (~t & s)                             31   Material Implication
33. p -> (~t & s)                          21,32   Hypothetical Syllogism
34. ~p v (~t & s)                             33   Material Implication
35. (~p v ~t) & (~p v s)                      34   Distribution
36. (~p v s) & (~p v ~t)                      35   Commutation
37. ~p v s                                    35   Simplification
38. p -> s                                    37   Material Implication
39. p -> ~p                                38,10   Hypothetical Syllogism
40. ~p v ~p                                   39   Material Implication
41. ~p                                        40   Tautology
42. s                                      14,41   Modus Ponens
43. ~t & s                                 29,42   Modus Ponens
44. ~t                                        43   Simplification
45  ~t & ~p                                44,41   Conjunction

1. (p <--> q) -> s
2. ~(~r -> t)
3. ~q v ~s Therefore: (t v p) -> (~t & ~ q)
--------------------------------------------
4.  ~(~~r v t)                               2     Material Implication
5.  ~(r v t)                                 4     Double Negation
6.   ~r & ~t                                 5     De Morgan's Law
7.   ~t & ~r                                 6     Commutation
8.   ~t                                      7     Simplification
---
9.  ~s v ~q                                  3     Commutation
10.  s -> ~q                                 9     Material Implication
---
11. (p <--> q) -> ~q                         1,10  Hypothetical Syllogism 
12. [(p & q) v (~p & ~q) ] -> ~q             11    Material Equivalence
13. ~[(p & q) v (~p & ~q) ] v ~q             12    Material Implication
14. [~(p & q) & ~(~p & ~q) ] v ~q            13    De Morgan's Law
15. ~q v [~(p & q) & ~(~p & ~q) ]            14    Commutation
16. [~q v (~p v ~q)] & [~q v (p v q) ]       15    Distribution
17. ~q v (~q v ~p)                           16    Simplification
18. (~q v ~q) v ~p                           17    Association
19. ~q v ~p                                  18    Tautology
20. ~p v ~q                                  19    Commutation
---
21. ~t & (~p v ~q)                           8,20  Conjunction
22. (~t & ~p) v (~t & ~q)                    21    Distribution
23. ~(t v p) v (~t & ~q)                     22    De Morgan's Law
24. (t v p) -> (~t & ~q)                     23    Material Implication

Note: try to involve every premise (the given lines) to find the conclusion.


1. p -> (q & r)
2. ~t -> (~r v ~s) Therefore: (p & s) -> t 
----------------------------------------------
3.  ~~t v (~r v ~s)                          2     Material Implication
4.  t v (~r v ~s)                            3     Double Negation
5.  (~r v ~s) v t                            4     Commutation
6.  ~r v (~s v t)                            5     Association
7.  ~r v (s -> t)                            6     Material Implication
8.  r -> (s -> t)                            7     Material Implication
9. ~p v (q & r)                              1     Material Implication
10. (~p v q) & (~p v r)                      9     Distribution
11. (~p v r) & (~p v q)                      10    Commutation
12. ~p v r                                   11    Simplification
13. p -> r                                   12    Material Implication
14. p -> (s -> t)                            13,8  Hypothetical Syllogism
15. (p & s) -> t                             14    Exportation

1. p -> (q <--> r)
2. ~q -> r
3. ~(q <--> s)
4. p -> s Therefore: ~p
------------------------
5.  ~[(q & s) v (~q & ~s)]                  3     Material Equivalence
6.  ~(q & s) & ~(~q & ~s)                   5     De Morgan's Law
7.  ~(~q & ~s) & ~(q & s)                   6     Commutation
8.  ~(q & s)                                6     Simplification     
9.  ~q v ~s                                 8     De Morgan's Law
10.  q -> ~s                                9     Material Implication
11.  ~~s -> ~q                             10     Transposition
12.  s -> ~q                               11     Double Negation
13.  s -> r                              12,2     Hypothetical Syllogism
14.  ~(~q & ~s)                             7     Simplification     
15.  ~~q v ~~s                             14     De Morgan's Law
16.  ~q -> ~~s                             15     Material Implication
17.  ~q -> s                               16     Double Negation
18.  ~q -> r                            17,13     Hypothetical Syllogism
--------------
19.  p -> [(q -> r) & (r -> q)]             1     Material Equivalence
20.  ~p v [(~q v r) & (~r v q)]            19     Material Implication
21.  [~p v (~q v r)] & [~p v (~r v q)]     20     Distribution
22.  [~p v (~r v q)] & [~p v (~q v r)]     21     Commutation
23.  ~p v (~r v q)                         22     Simplification
24.  p -> (~r v q)                         23     Material Implication
25.  p -> (r -> q)                         24     Material Implication
26.  (p & r) -> q                          25     Exportation
27.  (p & r) -> ~s                      26,10     Hypothetical Syllogism
28.  ~~s -> ~(p & r)                       27     Transposition
29.  s -> ~(p & r)                         28     Double Negation
30. ~q -> ~(p & r)                      17,29     Hypothetical Syllogism
31. q v (~p v ~r)                          30     Material Implication
32. (q v ~p) v ~r                          31     Association
33. ~r v (q v ~p)                          32     Commutation
34. ~r v (~p v q)                          33     Commutation
35. r -> (p -> q)                          34     Material Implication
36. ~q -> (p -> q)                      18,35     Hypothetical Syllogism  
37. (~q & p) -> q                          36     Exportation
38. ~(~q & p) v q                          37     Material Implication
39. (~~q v ~p) v q                         38     De Morgan's Law
40. (q v ~p) v q                           39     Double Negation
41. q v (q v ~p)                           40     Commutation
42. (q v q) v ~p                           41     Association
43. q v ~p                                 42     Tautology
44. ~p v q                                 43     Commutation
45. p -> q                                 44     Material Implication
46. ~s -> ~p                                4     Transposition
47. q -> ~p                             10,46     Hypothetical Syllogism
48. p -> ~p                             45,47     Hypothetical Syllogism
49. ~p v ~p                                48     Material Implication
50. ~p                                     49     Tautology

1. p <--> q
2. p -> r
3. p -> (r -> s) Therefore: q -> s
------------------------------------
4.  (p -> q) & (q -> p)                      3     Material Equivalence
5.  (q -> p) & (p -> q)                      4     Commutation
6.  q -> p                                   5     Simplification
7.  q -> r                                   6,2   Hypothetical Syllogism
8.  q -> (r -> s)                            6,3   Hypothetical Syllogism
9.  (q & r) -> s                             8     Exportation
10. ~(q & r) v s                             9     Material Implication    
11. (~q v ~r) v s                            10    DeMorgan's Theorem
12. ~q v (~r v s)                            11    Association
13. ~q v (s v ~r)                            12    Commutation
14. (~q v s) v ~r                            13    Association
15. (q -> s) v ~r                            14    Material Implication
16. ~r v (q -> s)                            15    Commutation
17. r -> (q -> s)                            16    Material Implication
18. q -> (q -> s)                            7,17  Hypothetical Syllogism
19. (q & q) -> s                             18    Exportation
20. q -> s                                   19    Tautology

1. p <--> q
2. p -> r
3. p -> (r -> s) Therefore: q -> s
------------------------------------
4.  (p -> q) & (q -> p)                      3     Material Equivalence
5.  (q -> p) & (p -> q)                      4     Commutation
6.  q -> p                                   5     Simplification
7.  q -> r                                   6,2   Hypothetical Syllogism
8.  q -> (r -> s)                            6,3   Hypothetical Syllogism
9.  (q & r) -> s                             8     Exportation
10. ~(q & r) v s                             9     Material Implication    
11. (~q v ~r) v s                            10    DeMorgan's Theorem
12. ~q v (~r v s)                            11    Association
13. ~q v (s v ~r)                            12    Commutation
14. (~q v s) v ~r                            13    Association
15. (q -> s) v ~r                            14    Material Implication
16. ~r v (q -> s)                            15    Commutation
17. r -> (q -> s)                            16    Material Implication
18. q -> (q -> s)                            7,17  Hypothetical Syllogism
19. (q & q) -> s                             18    Exportation
20. q -> s                                   19    Tautology

1. (p v q) -> r
2. r -> (s <--> t)
3. s -> ~t Therefore: ~p v ~t
-------------------------------
4.  r -> [ (s -> t) & (t -> s) ]                      2  Material Equivalence 
5.  ~r v [ (s -> t) & (t -> s) ]                      4  Material Implication
6.  [~r v (s -> t)] & [~r v (t -> s)]                 5  Distribution
7.  [~r v (t -> s)] & [~r v (s -> t)]                 6  Commutation
8.  ~r v (t -> s)                                     7  Simplification
9.  r -> (t -> s)                                     8  Material Implication
10. (r & t) -> s                                      9  Exportation
11. (r & t) -> ~t                                  3,10  Hypothetical Syllogism
12. r -> (t -> ~t)                                   11  Exportation
13. r -> (~t v ~t)                                   12  Material Implication 
14. r -> ~t                                          13  Tautology
15. (p v q) -> ~t                                  1,14  Hypothetical Syllogism
16. ~(p v q) v ~t                                    15  Material Implication
17. (~p & ~q) v ~t                                   16  DeMorgan's Theorem
18. ~t v (~p & ~q)                                   17  Commutation
19. (~t v ~p) & (~t v ~q)                            18  Distribution
20. ~t v ~p                                          19  Simplification
21. ~p v ~t                                          20  Commutation