An exercise in logical equivalency

Double Negation

Nc3 is the same as not not Nc3

Duplication

Eliminates redundancy

Nc3 is the same as (Nc3 or Nc3)
Nc3 is the same as (Nc3 and Nc3)

Commutation

(Nc3 or d4) is the same as (d4 or Nc3)
(Nc3 and d4) is the same as (d4 and Nc3)

Association

(Nc3 or (d4 or e4)) is the same as ((Nc3 or d4) or e4)
((Nc3 and d4) and e4) is the same as (Nc3 and (d4 and e4))

Contraposition

If Nc3 then Bb4 is the same as If not Nc3 then not Bb4

De Morgan's

Not Bb4 or Nc6 is the same as Not Bb4 and not Nc6
Not Bb4 and Nc6 is the same as Not Bb4 or not Nc6

Biconditional Exchange

Bb4 if and only if Nc3 is the same as ((If Bb4 then Nc3) and (If Nc3 then Bb4))

Conditional Exchange

If Nc3 then Bb4 is the same as Not Nc3 or Bb4

Distribution

(Nc3 and (Bb4 or Nc6)) is the same as ((Nc3 and Bb4) or (Nc3 and Nc6))

Exportation

(If Nc3 then (if Bb4 then e4)) is the same as ((If Nc3 and Bb4) then e4)

Note: These replacement rules work in both directions