YES

Problem:
 and(P,forall()) -> forall()
 or(P,forall()) -> forall()
 and(forall(),P) -> forall()
 or(forall(),P) -> forall()
 not(forall()) -> exists()
 and(P,exists()) -> exists()
 or(P,exists()) -> exists()
 and(exists(),P) -> exists()
 or(exists(),P) -> exists()
 not(exists()) -> forall()

Proof:
 Matrix Interpretation Processor: dim=3
  
  interpretation:
              [0]
   [exists] = [0]
              [0],
   
               [1 0 0]     [1]
   [not](x0) = [0 0 0]x0 + [0]
               [0 0 0]     [0],
   
                  [1 0 0]     [1 0 0]     [1]
   [or](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0]
                  [0 0 0]     [0 0 0]     [0],
   
                   [1 0 0]     [1 0 0]     [1]
   [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0]
                   [0 0 0]     [0 0 0]     [0],
   
              [0]
   [forall] = [0]
              [0]
  orientation:
                     [1 0 0]    [1]    [0]           
   and(P,forall()) = [0 0 0]P + [0] >= [0] = forall()
                     [0 0 0]    [0]    [0]           
   
                    [1 0 0]    [1]    [0]           
   or(P,forall()) = [0 0 0]P + [0] >= [0] = forall()
                    [0 0 0]    [0]    [0]           
   
                     [1 0 0]    [1]    [0]           
   and(forall(),P) = [0 0 0]P + [0] >= [0] = forall()
                     [0 0 0]    [0]    [0]           
   
                    [1 0 0]    [1]    [0]           
   or(forall(),P) = [0 0 0]P + [0] >= [0] = forall()
                    [0 0 0]    [0]    [0]           
   
                   [1]    [0]           
   not(forall()) = [0] >= [0] = exists()
                   [0]    [0]           
   
                     [1 0 0]    [1]    [0]           
   and(P,exists()) = [0 0 0]P + [0] >= [0] = exists()
                     [0 0 0]    [0]    [0]           
   
                    [1 0 0]    [1]    [0]           
   or(P,exists()) = [0 0 0]P + [0] >= [0] = exists()
                    [0 0 0]    [0]    [0]           
   
                     [1 0 0]    [1]    [0]           
   and(exists(),P) = [0 0 0]P + [0] >= [0] = exists()
                     [0 0 0]    [0]    [0]           
   
                    [1 0 0]    [1]    [0]           
   or(exists(),P) = [0 0 0]P + [0] >= [0] = exists()
                    [0 0 0]    [0]    [0]           
   
                   [1]    [0]           
   not(exists()) = [0] >= [0] = forall()
                   [0]    [0]           
  problem:
   
  Qed