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
  Ex(x,All(y,p)) => All(y,Ex(x,p)) is valid
proof
  not x in still_not-bound_in Ex(x,p) by Th6;
  then
A1: not x in still_not-bound_in All(y,Ex(x,p)) by Th5;
  All(y,p => Ex(x,p)) is valid & All(y,p => Ex(x,p)) => (All(y,p) => All(y
  ,Ex( x,p))) is valid by Th15,Th23,Th30;
  then All(y,p) => All(y,Ex(x,p)) is valid by CQC_THE1:65;
  hence thesis by A1,Th19;
end;
