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 Th61:
  Ex(x,p => q) => (All(x,p) => Ex(x,q)) is valid & (All(x,p) => Ex
  (x,q)) => Ex(x,p => q) is valid
proof
  All(x,p) => p is valid by CQC_THE1:66;
  then
A1: (p => q) => (All(x,p) => q) is valid by LUKASI_1:41;
  ( not x in still_not-bound_in All(x,p))& not x in still_not-bound_in Ex(
  x, q ) by Th5,Th6;
  then
A2: not x in still_not-bound_in All(x,p) => Ex(x,q) by Th7;
  q => Ex(x,q) is valid by Th15;
  then (p => q) => (All(x,p) => Ex(x,q)) is valid by A1,Lm16;
  hence Ex(x,p => q) => (All(x,p) => Ex(x,q)) is valid by A2,Th19;
  All(x,p '&' 'not' q) => (All(x,p) '&' All(x,'not' q)) is valid by Th36;
  then
A3: 'not'(All(x,p) '&' All(x,'not' q)) => 'not' All(x,p '&' 'not' q) is
  valid by LUKASI_1:52;
  'not' All(x,p '&' 'not' q) => Ex(x,'not'(p '&' 'not' q)) is valid by Th51;
  then 'not'(All(x,p) '&' All(x,'not' q)) => Ex(x,'not'(p '&' 'not' q)) is
  valid by A3,LUKASI_1:42;
  then
A4: 'not'(All(x,p) '&' All(x,'not' q)) => Ex(x,p => q) is valid by
QC_LANG2:def 2;
  All(x,'not' q) => 'not' 'not' All(x,'not' q) is valid;
  then
A5: (All(x,p) '&' All(x,'not' q)) => (All(x,p) '&' 'not' 'not' All(x,'not'
  q)) is valid by Lm9;
  All(x,p) => Ex(x,q) = All(x,p) => 'not' All(x,'not' q) by QC_LANG2:def 5
    .= 'not'(All(x,p) '&' 'not' 'not' All(x,'not' q)) by QC_LANG2:def 2;
  then
  (All(x,p) => Ex(x,q)) => 'not'(All(x,p) '&' All(x,'not' q)) is valid by A5,
LUKASI_1:52;
  hence thesis by A4,LUKASI_1:42;
end;
