reserve A for QC-alphabet;
reserve X,T for Subset of CQC-WFF(A);
reserve F,G,H,p,q,r,t for Element of CQC-WFF(A);
reserve s,h for QC-formula of A;
reserve x,y for bound_QC-variable of A;
reserve f for FinSequence of [:CQC-WFF(A),Proof_Step_Kinds:];
reserve i,j for Element of NAT;

theorem Th23:
  p is valid implies All(x,p) is valid
proof
A1: p => ((All(x,p) => All(x,p)) => p) is valid;
  not x in still_not-bound_in All(x,p) by Th5;
  then
A2: not x in still_not-bound_in All(x,p) => All(x,p) by Th7;
  assume p is valid;
  then (All(x,p) => All(x,p)) => p is valid by A1,CQC_THE1:65;
  then (All(x,p) => All(x,p)) => All(x,p) is valid by A2,CQC_THE1:67;
  hence thesis by CQC_THE1:65;
end;
