Removing the first index along N-1 dimensions of an Array

General Usage
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2

can you explain the need in more detail?

Does this come close to what you need?

m=rand(1:100, 4,3,5)

f(c)=!(c.I[1]==1 &&  c.I[2]==1) &&
    !(c.I[2]==1 &&  c.I[3]==1) &&
    !(c.I[1]==1 &&  c.I[3]==1)

m[filter(f, CartesianIndices(size(m)))]

more generally, the following predicate could be used (it remains to clarify / specify the case i = j = k = … = 1)


f(c)=sum((==(1)).(c.I))!=length(c.I)-1