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:
 DP Processor:
  DPs:
   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,y) -> not#(y)
   =#(x,y) -> if#(x,y,not(y))
   =#(x,y) -> if#(y,false(),true())
   =#(x,y) -> if#(x,y,if(y,false(),true()))
  TRS:
   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()))
  TDG Processor:
   DPs:
    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,y) -> not#(y)
    =#(x,y) -> if#(x,y,not(y))
    =#(x,y) -> if#(y,false(),true())
    =#(x,y) -> if#(x,y,if(y,false(),true()))
   TRS:
    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()))
   graph:
    =#(x,y) -> not#(y) -> not#(x) -> if#(x,false(),true())
   SCC Processor:
    #sccs: 0
    #rules: 0
    #arcs: 1/64