reserve Al for QC-alphabet;
reserve a,b,c,d for object,
  i,j,k,m,n for Nat,
  p,q,r for Element of CQC-WFF(Al),
  x,y,y0 for bound_QC-variable of Al,
  X for Subset of CQC-WFF(Al),
  A for non empty set,
  J for interpretation of Al,A,
  v,w for Element of Valuations_in(Al,A),
  Sub for CQC_Substitution of Al,
  f,f1,g,h,h1 for FinSequence of CQC-WFF(Al);
reserve fin,fin1 for FinSequence;
reserve PR,PR1 for FinSequence of [:set_of_CQC-WFF-seq(Al),Proof_Step_Kinds:];
reserve a for Element of A;

theorem Th29:
  Suc(f) = p.(x,y) & Ant(f) |= Suc(f) & not y in
still_not-bound_in Ant(f) & not y in still_not-bound_in All(x,p) implies Ant(f)
  |= All(x,p)
proof
  assume that
A1: Suc(f) = p.(x,y) & Ant(f) |= Suc(f) and
A2: not y in still_not-bound_in (Ant(f)) and
A3: not y in still_not-bound_in All(x,p);
  let A,J,v such that
A4: J,v |= Ant(f);
  for a holds J,v.(x|a) |= p
  proof
    let a;
    v.(y|a)|still_not-bound_in (Ant(f)) = v|still_not-bound_in (Ant(f)) by A2
,Th26;
    then J,v.(y|a) |= Ant(f) by A4,Th27;
    then J,v.(y|a) |= p.(x,y) by A1;
    then
    ex a1 being Element of A st v.(y|a).y = a1 & J,v.(y|a).(x |a1) |= p by Th24
;
    then
A5: J,v.(y|a).(x|a) |= p by SUBLEMMA:49;
    v.(y|a).(x|a)|still_not-bound_in p = v.(x|a)|still_not-bound_in p by A3
,Th28;
    hence thesis by A5,SUBLEMMA:68;
  end;
  hence thesis by SUBLEMMA:50;
end;
