YES

Problem:
 not(x) -> if(x,false(),true())
 and(x,y) -> if(x,y,false())
 or(x,y) -> if(x,true(),y)
 implies(x,y) -> if(x,y,true())
 =(x,x) -> true()
 =(x,y) -> if(x,y,not(y))
 if(true(),x,y) -> x
 if(false(),x,y) -> y
 if(x,x,if(x,false(),true())) -> true()
 =(x,y) -> if(x,y,if(y,false(),true()))

Proof:
 Matrix Interpretation Processor: dim=1
  
  interpretation:
   [=](x0, x1) = 4x0 + 7x1 + 7,
   
   [implies](x0, x1) = 4x0 + 4x1 + 4,
   
   [or](x0, x1) = 4x0 + 7x1 + 1,
   
   [and](x0, x1) = 4x0 + 2x1 + 6,
   
   [if](x0, x1, x2) = x0 + x1 + 6x2,
   
   [true] = 0,
   
   [false] = 1,
   
   [not](x0) = x0 + 1
  orientation:
   not(x) = x + 1 >= x + 1 = if(x,false(),true())
   
   and(x,y) = 4x + 2y + 6 >= x + y + 6 = if(x,y,false())
   
   or(x,y) = 4x + 7y + 1 >= x + 6y = if(x,true(),y)
   
   implies(x,y) = 4x + 4y + 4 >= x + y = if(x,y,true())
   
   =(x,x) = 11x + 7 >= 0 = true()
   
   =(x,y) = 4x + 7y + 7 >= x + 7y + 6 = if(x,y,not(y))
   
   if(true(),x,y) = x + 6y >= x = x
   
   if(false(),x,y) = x + 6y + 1 >= y = y
   
   if(x,x,if(x,false(),true())) = 8x + 6 >= 0 = true()
   
   =(x,y) = 4x + 7y + 7 >= x + 7y + 6 = if(x,y,if(y,false(),true()))
  problem:
   not(x) -> if(x,false(),true())
   and(x,y) -> if(x,y,false())
   if(true(),x,y) -> x
  Matrix Interpretation Processor: dim=3
   
   interpretation:
                    [1 0 0]          [1]
    [and](x0, x1) = [0 0 0]x0 + x1 + [0]
                    [0 0 0]          [0],
    
                       [1 0 0]          [1 0 0]  
    [if](x0, x1, x2) = [0 0 0]x0 + x1 + [0 0 0]x2
                       [0 0 0]          [0 0 0]  ,
    
             [0]
    [true] = [0]
             [0],
    
              [0]
    [false] = [0]
              [0],
    
                [1 0 0]     [1]
    [not](x0) = [0 0 0]x0 + [0]
                [0 0 0]     [0]
   orientation:
             [1 0 0]    [1]    [1 0 0]                        
    not(x) = [0 0 0]x + [0] >= [0 0 0]x = if(x,false(),true())
             [0 0 0]    [0]    [0 0 0]                        
    
               [1 0 0]        [1]    [1 0 0]                       
    and(x,y) = [0 0 0]x + y + [0] >= [0 0 0]x + y = if(x,y,false())
               [0 0 0]        [0]    [0 0 0]                       
    
                         [1 0 0]          
    if(true(),x,y) = x + [0 0 0]y >= x = x
                         [0 0 0]          
   problem:
    if(true(),x,y) -> x
   Matrix Interpretation Processor: dim=3
    
    interpretation:
                        [1 0 1]     [1 1 1]     [1 0 0]     [0]
     [if](x0, x1, x2) = [0 0 0]x0 + [1 1 1]x1 + [0 0 0]x2 + [1]
                        [0 0 0]     [1 1 1]     [0 0 0]     [1],
     
              [0]
     [true] = [0]
              [1]
    orientation:
                      [1 1 1]    [1 0 0]    [1]         
     if(true(),x,y) = [1 1 1]x + [0 0 0]y + [1] >= x = x
                      [1 1 1]    [0 0 0]    [1]         
    problem:
     
    Qed