reserve Y for non empty set;

theorem
  for a,b being Function of Y,BOOLEAN, G being Subset of
  PARTITIONS(Y), PA being a_partition of Y
   holds All(a 'eqv' b,PA,G) = All(a 'imp' b,PA,G) '&' All(b 'imp' a,PA,G)
proof
  let a,b be Function of Y,BOOLEAN;
  let G be Subset of PARTITIONS(Y);
  let PA be a_partition of Y;
    let z be Element of Y;
    All(a 'eqv' b,PA,G).z =(All((a 'imp' b) '&' (b 'imp' a),PA,G)).z by Th7
      .=(All(a 'imp' b,PA,G) '&' All(b 'imp' a,PA,G)).z by BVFUNC_1:39;
    hence thesis;
end;
