YES

Problem:
 and(not(not(x)),y,not(z)) -> and(y,band(x,z),x)

Proof:
 Matrix Interpretation Processor: dim=1
  
  interpretation:
   [and](x0, x1, x2) = x0 + x1 + 4x2 + 2,
   
   [not](x0) = 4x0 + 1,
   
   [band](x0, x1) = x0 + x1
  orientation:
   and(not(not(x)),y,not(z)) = 16x + y + 16z + 11 >= 5x + y + z + 2 = and(y,band(x,z),x)
  problem:
   
  Qed