Symmetric difference of set identity

[el_llavero]

Is there a shorter way to verify this identity, as you can see I haven't even finished it. I know you can use Ven diagrams and truth tables but I wanted to avoid them inorder to use a more general formal approach. picture is attached

 

[el_llavero]

Sets are associative and commutative over the symmetric difference operator, use this to combine like terms, then simplify. Similar technique can be used on variations of this identity to build your own identities which can be used as short cuts in other proofs containing the symmetric difference operator.

see attachment