To show that A∩B∩C=A∪B∪C
Let x∈A∪B∪C
x∈U and x∈/A∩B∩Cx∈U and x∈/A or x∈/B or x∈/Cx∈U and x∈/A or x∈U and x∈/B or x∈U and x∈/Cx∈A or x∈B or x∈Cx∈A∪B∪C⟹A∪B∪C⊆A∪B∪C
Conversely,
Let x∈A∪B∪C
x∈A or x∈B or x∈Cx∈U and x∈/A or x∈U and x∈/B or x∈U and x∈/Cx∈U and x∈/A or x∈/B or x∈/Cx∈U and x∈/A∩B∩Cx∈A∪B∪C⟹A∪B∪C⊆A∪B∪C
Hence,
A∩B∩C=A∪B∪C