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 Th78:
  (All(x,p) => q) => Ex(x,p => q) is valid
proof
  All(x,p '&' 'not' q) => All(x,p) '&' 'not' q is valid by Th65;
  then 'not'(All(x,p) '&' 'not' q) => 'not' All(x,p '&' 'not' q) is valid by
LUKASI_1:52;
  then
A1: (All(x,p) => q) => 'not' All(x,p '&' 'not' q) is valid by QC_LANG2:def 2;
  All(x,'not' 'not'(p '&' 'not' q) => (p '&' 'not' q)) is valid & All(x,
  'not' 'not'(p '&' 'not' q) => (p '&' 'not' q)) => (All(x,'not' 'not'(p '&'
  'not' q)) => All(x,(p '&' 'not' q))) is valid by Th23,Th30;
  then
  All(x,'not' 'not'(p '&' 'not' q)) => All(x,(p '&' 'not' q)) is valid by
CQC_THE1:65;
  then
  'not' All(x,p '&' 'not' q) => 'not' All(x,'not' 'not'(p '&' 'not' q)) is
  valid by LUKASI_1:52;
  then (All(x,p) => q) => 'not' All(x,'not' 'not'(p '&' 'not' q)) is valid by
A1,LUKASI_1:42;
  then (All(x,p) => q) => Ex(x,'not'(p '&' 'not' q)) is valid by QC_LANG2:def 5
;
  hence thesis by QC_LANG2:def 2;
end;
