Distant is the opposite of Local. Local is related to Place. Place is related to Distance. Distance is related to Closer. 
Farmer is related to Barn. Barn is related to Pig. 
Distant <--Antonym--> Local <--RelatedTo--> Place <--RelatedTo--> Distance <--RelatedTo--> Closer 
Farmer <--RelatedTo--> Barn <--RelatedTo--> Pig 