I.   Use IP in derivations for the following problems:

           A.   Derive P v ~P from scratch

               1.       ~(P v ~P)               Assumption
               2.       ~P & ~~P                1, DM
               3.   P v ~P                      1-2, IP

           B.   Derive P -> P from scratch

               1.       ~(P -> P)               Assumption
               2.       ~(~P v P)               1, Impl.
               3.       ~~P & ~P                2, DM
               4.   P -> P                      1-3, IP

           C.   Derive ~(P -> Q) from

               1.   P & ~Q                      Premise
               2.       P -> Q                  Assumption
               3.       ~Q & P                  1, Comm.
               4.       ~Q                      3, Simp.
               5.       ~P                      2, 4, MT
               6.       P                       1, Simp.
               7.       P & ~P                  5, 6, Conj.
               8.   ~(P -> Q)                   2-7, IP

           D.   Without using De Morgan's Theorems (DM), derive ~(P v Q)
                from

               1.   ~P                          Premise
               2.   ~Q                          Premise
               3.       P v Q                   Assumption
               4.       Q                       1, 3, DS
               5.       Q & ~Q                  2, 4, Conj.
               6.   ~(P v Q)                    3-5, IP

           E.   Without using De Morgan's Theorems (DM), derive (~P v ~Q)
                -> ~(P & Q) from scratch. (Hint: use an in indirect proof
                inside of a conditional proof.)

               1.       ~P v ~Q                 Assumption
               2.           P & Q               Assumption
               3.           P                   2, Simp.
               4.           ~~P                 3, DN
               5.           ~Q                  1, 4, DS
               6.           Q & P               2, Comm.
               7.           Q                   6, Simp.
               8.           Q & ~Q              5, 7, Conj.
               9.       ~(P & Q)                2-8, IP
              10.   (~P v ~Q) -> ~(P & Q)       1-9, CP