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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
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
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