When you take G the equality, then your list of axioms is that of a commutative group. The thing you need to prove is that the identity element is unique.
Take two identity's c and c', then
3) G(F(c',c),c')
and
3) G(F(c,c'),c)
and
4) G(F(c,c'),F(c',c))
So by (5), we get that G(c,c')
But that doesn't give equality, however...
