reserve Y for non empty set,
  a,b,c,d for Function of Y,BOOLEAN;
reserve Y for non empty set,
  a,b,c for Function of Y,BOOLEAN;

theorem
  a 'eqv' (a 'nand' b) = a '&' 'not' b
proof
  thus a 'eqv' (a 'nand' b) =(a '&' 'not' (a '&' b)) 'or' ('not' a '&' a '&' b
  ) by Th53
    .=(a '&' 'not' (a '&' b)) 'or' (O_el(Y) '&' b) by BVFUNC_4:5
    .=(a '&' 'not' (a '&' b)) 'or' O_el(Y) by BVFUNC_1:5
    .=(a '&' 'not' (a '&' b)) by BVFUNC_1:9
    .=(a '&' ('not' a 'or' 'not' b)) by BVFUNC_1:14
    .=(a '&' 'not' a) 'or' (a '&' 'not' b) by BVFUNC_1:12
    .=O_el(Y) 'or' (a '&' 'not' b) by BVFUNC_4:5
    .=a '&' 'not' b by BVFUNC_1:9;
end;
