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 Th71:
  not x in still_not-bound_in p implies (p '&' Ex(x,q)) => Ex(x,p
  '&' q) is valid & Ex(x,p '&' q) => (p '&' Ex(x,q)) is valid
proof
  assume
A1: not x in still_not-bound_in p;
  p '&' q => Ex(x,p '&' q) is valid by Th15;
  then
A2: q => (p => Ex(x,p '&' q)) is valid by Th4;
  not x in still_not-bound_in Ex(x,p '&' q) by Th6;
  then not x in still_not-bound_in p => Ex(x,p '&' q) by A1,Th7;
  then Ex(x,q) => (p => Ex(x,p '&' q)) is valid by A2,Th19;
  hence (p '&' Ex(x,q)) => Ex(x,p '&' q) is valid by Th2;
  q => Ex(x,q) is valid by Th15;
  then
A3: p '&' q => p '&' Ex(x,q) is valid by Lm9;
  not x in still_not-bound_in Ex(x,q) by Th6;
  then not x in still_not-bound_in p '&' Ex(x,q) by A1,Th8;
  hence thesis by A3,Th19;
end;
