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:
   [and](x0, x1) = x0 + x1,
   
   [false] = 0,
   
   [if](x0, x1, x2) = x0 + x1 + 2x2,
   
   [or](x0, x1) = x0 + 2x1,
   
   [not](x0) = x0 + 1,
   
   [=](x0, x1) = 5x0 + 4x1 + 2,
   
   [implies](x0, x1) = 4x0 + 4x1,
   
   [true] = 0
  orientation:
   not(x) = x + 1 >= x = if(x,false(),true())
   
   and(x,y) = x + y >= x + y = if(x,y,false())
   
   or(x,y) = x + 2y >= x + 2y = if(x,true(),y)
   
   implies(x,y) = 4x + 4y >= x + y = if(x,y,true())
   
   =(x,x) = 9x + 2 >= 0 = true()
   
   =(x,y) = 5x + 4y + 2 >= x + 3y + 2 = if(x,y,not(y))
   
   if(true(),x,y) = x + 2y >= x = x
   
   if(false(),x,y) = x + 2y >= y = y
   
   if(x,x,if(x,false(),true())) = 4x >= 0 = true()
   
   =(x,y) = 5x + 4y + 2 >= x + 3y = if(x,y,if(y,false(),true()))
  problem:
   and(x,y) -> if(x,y,false())
   or(x,y) -> if(x,true(),y)
   implies(x,y) -> if(x,y,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()
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [and](x0, x1) = 4x0 + x1 + 4,
    
    [false] = 0,
    
    [if](x0, x1, x2) = 4x0 + x1 + 4x2,
    
    [or](x0, x1) = 4x0 + 4x1,
    
    [not](x0) = x0,
    
    [=](x0, x1) = 4x0 + 5x1,
    
    [implies](x0, x1) = 5x0 + x1,
    
    [true] = 0
   orientation:
    and(x,y) = 4x + y + 4 >= 4x + y = if(x,y,false())
    
    or(x,y) = 4x + 4y >= 4x + 4y = if(x,true(),y)
    
    implies(x,y) = 5x + y >= 4x + y = if(x,y,true())
    
    =(x,y) = 4x + 5y >= 4x + 5y = if(x,y,not(y))
    
    if(true(),x,y) = x + 4y >= x = x
    
    if(false(),x,y) = x + 4y >= y = y
    
    if(x,x,if(x,false(),true())) = 21x >= 0 = true()
   problem:
    or(x,y) -> if(x,true(),y)
    implies(x,y) -> if(x,y,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()
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [false] = 0,
     
     [if](x0, x1, x2) = 2x0 + 3x1 + 4x2 + 1,
     
     [or](x0, x1) = 4x0 + 6x1 + 6,
     
     [not](x0) = x0,
     
     [=](x0, x1) = 2x0 + 7x1 + 1,
     
     [implies](x0, x1) = 3x0 + 3x1 + 5,
     
     [true] = 1
    orientation:
     or(x,y) = 4x + 6y + 6 >= 2x + 4y + 4 = if(x,true(),y)
     
     implies(x,y) = 3x + 3y + 5 >= 2x + 3y + 5 = if(x,y,true())
     
     =(x,y) = 2x + 7y + 1 >= 2x + 7y + 1 = if(x,y,not(y))
     
     if(true(),x,y) = 3x + 4y + 3 >= x = x
     
     if(false(),x,y) = 3x + 4y + 1 >= y = y
     
     if(x,x,if(x,false(),true())) = 13x + 21 >= 1 = true()
    problem:
     implies(x,y) -> if(x,y,true())
     =(x,y) -> if(x,y,not(y))
    Matrix Interpretation Processor: dim=2
     
     interpretation:
                         [1 0]     [1 1]     [1 1]     [0]
      [if](x0, x1, x2) = [0 0]x0 + [2 1]x1 + [1 0]x2 + [1],
      
                  [1 1]     [1]
      [not](x0) = [1 0]x0 + [1],
      
                    [2 2]     [3 2]     [2]
      [=](x0, x1) = [0 0]x0 + [3 2]x1 + [2],
      
                          [1 1]     [2 2]     [3]
      [implies](x0, x1) = [1 1]x0 + [2 1]x1 + [1],
      
               [0]
      [true] = [2]
     orientation:
                     [1 1]    [2 2]    [3]    [1 0]    [1 1]    [2]                 
      implies(x,y) = [1 1]x + [2 1]y + [1] >= [0 0]x + [2 1]y + [1] = if(x,y,true())
      
               [2 2]    [3 2]    [2]    [1 0]    [3 2]    [2]                 
      =(x,y) = [0 0]x + [3 2]y + [2] >= [0 0]x + [3 2]y + [2] = if(x,y,not(y))
     problem:
      =(x,y) -> if(x,y,not(y))
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [if](x0, x1, x2) = 2x0 + 2x1 + x2,
       
       [not](x0) = 4x0,
       
       [=](x0, x1) = 2x0 + 6x1 + 1
      orientation:
       =(x,y) = 2x + 6y + 1 >= 2x + 6y = if(x,y,not(y))
      problem:
       
      Qed