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 Th74:
  not x in still_not-bound_in p implies All(x,p => q) => (p => All
  (x,q)) is valid & (p => All(x,q)) => All(x,p => q) is valid
proof
  assume
A1: not x in still_not-bound_in p;
  hence All(x,p => q) => (p => All(x,q)) is valid by Lm17;
  not x in still_not-bound_in All(x,q) by Th5;
  then not x in still_not-bound_in p => All(x,q) by A1,Th7;
  then
A2: All(x,(p => All(x,q)) => (p => q)) => ((p => All(x,q)) => All(x,p => q)
  ) is valid by Lm17;
  All(x,(All(x,q) => q) => ((p => All(x,q)) => (p => q))) is valid & All(x
,( All(x,q) => q) => ((p => All(x,q)) => (p => q))) => (All(x,All(x,q) => q) =>
  All(x,(p => All(x,q)) => (p => q))) is valid by Th23,Th30;
  then
A3: All(x,All(x,q) => q) => All(x,(p => All(x,q)) => (p => q)) is valid by
CQC_THE1:65;
  All(x,All(x,q) => q) is valid by Th23,CQC_THE1:66;
  then All(x,(p => All(x,q)) => (p => q)) is valid by A3,CQC_THE1:65;
  hence thesis by A2,CQC_THE1:65;
end;
