reserve Al for QC-alphabet;
reserve i,j,k for Nat,
  A,D for non empty set;
reserve f1,f2 for Element of Funcs(Valuations_in(Al,A),BOOLEAN),
  x,x1,y for bound_QC-variable of Al,
  v,v1 for Element of Valuations_in(Al,A);
reserve ll for CQC-variable_list of k,Al;
reserve p,q,s,t for Element of CQC-WFF(Al),
  J for interpretation of Al,A,
  P for QC-pred_symbol of k,Al,
  r for Element of relations_on A;
reserve u,w,z for Element of BOOLEAN;
reserve w,v2 for Element of Valuations_in(Al,A),
  z for bound_QC-variable of Al;
reserve u,w for Element of Valuations_in(Al,A);
reserve s9 for QC-formula of Al;

theorem
  J |= p => q & not x in still_not-bound_in p implies J |= p => All(x,q)
proof
  assume that
A1: for v holds J,v |= p => q and
A2: not x in still_not-bound_in p;
  let u;
  now
    assume
A3: J,u |= p;
    now
      let w;
      assume for y st x<>y holds w.y = u.y;
      then
A4:   J,w |= p by A2,A3,Th28;
      J,w |= p => q by A1;
      hence J,w |= q by A4,Th24;
    end;
    hence J,u |= All(x,q) by Th29;
  end;
  hence thesis by Th24;
end;
