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 Th68:
  not x in still_not-bound_in p implies (p 'or' All(x,q)) => All(x
  ,p 'or' q) is valid & All(x,p 'or' q) => (p 'or' All(x,q)) is valid
proof
A1: not x in still_not-bound_in All(x,p 'or' q) by Th5;
  All(x,p 'or' q) => (p 'or' q) is valid & (p 'or' q) => ('not' p => q) is
  valid by Lm11,CQC_THE1:66;
  then All(x,p 'or' q) => ('not' p => q) is valid by LUKASI_1:42;
  then
A2: (All(x,p 'or' q) '&' 'not' p) => q is valid by Th1;
  assume
A3: not x in still_not-bound_in p;
  then not x in still_not-bound_in 'not' p by QC_LANG3:7;
  then not x in still_not-bound_in All(x,p 'or' q) '&' 'not' p by A1,Th8;
  then (All(x,p 'or' q) '&' 'not' p) => All(x,q) is valid by A2,CQC_THE1:67;
  then
A4: All(x, p 'or' q) => ('not' p => All(x,q)) is valid by Th3;
  p => p is valid;
  then p => All(x,p) is valid by A3,CQC_THE1:67;
  then
A5: (p 'or' All(x,q)) => (All(x,p) 'or' All(x,q)) is valid by Lm10;
  (All(x,p) 'or' All(x,q)) => All(x,p 'or' q) is valid by Th39;
  hence (p 'or' All(x,q)) => All(x,p 'or' q) is valid by A5,LUKASI_1:42;
  ('not' p => All(x,q)) => (p 'or' All(x,q)) is valid by Lm12;
  hence thesis by A4,LUKASI_1:42;
end;
