Question 870562
<pre>
1.  M <-> N
2.  ~M v ~N
3.  O -> M
/~(N v O)
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4.  (M -> N) & (N -> M)                    1        ME
5.  (N -> M) & (M -> N)                    4        Comm
6.  N -> M                                 5        Simp
7.  ~~M -> ~N                              2        MI
8.  M -> ~N                                7        DN
9.  N -> ~N                                6,8      HS
10. ~~N -> ~N                              9        DN
11. ~N v ~N                                10       MI
12. ~N                                     11       Taut
13. M -> N                                 4        Simp
14. M -> ~~N                               13       DN
15. ~M                                     14,12    MT
16. O -> ~~M                               3        DN
17. ~O                                     16,15    MT
18. ~N & ~O                                12,17    Conj
19. ~(~~N v ~~O)                           18       DM
20. ~(N v O)                               19       DN
</pre>


Abbreviations/Acronyms Used:


Comm = Commutation
Simp = Simplification
MT = Modus Tollens
Taut = Tautology
Conj = Conjunction
DN = Double Negation
ME = Material Equivalence
MI = Material Implication
HS = Hypothetical Syllogism
DM = De Morgan's Law