reserve Al for QC-alphabet;
reserve a,b,c,d for object,
  i,k,n for Nat,
  p,q for Element of CQC-WFF(Al),
  x,y,y1 for bound_QC-variable of Al,
  A for non empty set,
  J for interpretation of Al,A,
  v,w for Element of Valuations_in(Al,A),
  f,g for Function,
  P,P9 for QC-pred_symbol of k,Al,
  ll,ll9 for CQC-variable_list of k,Al,
  l1 for FinSequence of QC-variables(Al),
  Sub,Sub9,Sub1 for CQC_Substitution of Al,
  S,S9,S1,S2 for Element of CQC-Sub-WFF(Al),
  s for QC-symbol of Al;
reserve vS,vS1,vS2 for Val_Sub of A,Al;
reserve B for Element of [:QC-Sub-WFF(Al),bound_QC-variables(Al):],
  SQ for second_Q_comp of B;
reserve B for CQC-WFF-like Element of [:QC-Sub-WFF(Al),
  bound_QC-variables(Al):],
  xSQ for second_Q_comp of [S,x],
  SQ for second_Q_comp of B;
reserve B1 for Element of [:QC-Sub-WFF(Al),bound_QC-variables(Al):];
reserve SQ1 for second_Q_comp of B1;
reserve a for Element of A;

theorem Th60:
  for v,w holds (v|still_not-bound_in (P!ll) = w|
  still_not-bound_in (P!ll) implies (J,v |= P!ll iff J,w |= P!ll))
proof
  let v,w;
  assume
A1: v|still_not-bound_in (P!ll) = w|still_not-bound_in (P!ll);
A2: still_not-bound_in (P!ll) = still_not-bound_in ll by QC_LANG3:5;
A3: w*'ll in J.P iff Valid(P!ll,J).w = TRUE by VALUAT_1:7;
A4: Valid(P!ll,J).v = TRUE iff v*'ll in J.P by VALUAT_1:7;
  ll*(w|still_not-bound_in ll) in J.P iff w*'ll in J.P by Th59;
  hence thesis by A1,A2,A4,A3,Th59,VALUAT_1:def 7;
end;
