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 Th19:
  p => q is valid & not x in still_not-bound_in q implies Ex(x,p) => q is valid
proof
  assume p => q is valid & not x in still_not-bound_in q;
  then 'not' q => 'not' p is valid & not x in still_not-bound_in 'not' q by
LUKASI_1:52,QC_LANG3:7;
  then 'not' q => All(x,'not' p) is valid by CQC_THE1:67;
  then 'not' All(x,'not' p) => 'not' 'not' q is valid by LUKASI_1:52;
  then Ex(x,p) => 'not' 'not' q is valid by QC_LANG2:def 5;
  hence thesis by LUKASI_1:55;
end;
