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 Th84:
  (p => Ex(x,q)) => Ex(x,p => q) is valid
proof
  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
A1: 'not' All(x,p '&' 'not' q) => 'not' All(x,'not' 'not'(p '&' 'not' q))
  is valid by LUKASI_1:52;
  All(x,'not' q) <=> 'not' Ex(x,q) is valid by Th54;
  then All(x,'not' q) => 'not' Ex(x,q) is valid by Lm14;
  then (p '&' All(x,'not' q)) => (p '&' 'not' Ex(x,q)) is valid by Lm9;
  then
A2: 'not'(p '&' 'not' Ex(x,q)) => 'not'(p '&' All(x,'not' q)) is valid by
LUKASI_1:52;
  All(x,p '&' 'not' q) => (p '&' All(x,'not' q)) is valid by Th64;
  then 'not'(p '&' All(x,'not' q)) => 'not' All(x,p '&' 'not' q) is valid by
LUKASI_1:52;
  then 'not'(p '&' 'not' Ex(x,q)) => 'not' All(x,p '&' 'not' q) is valid by A2,
LUKASI_1:42;
  then (p => Ex(x,q)) => 'not' All(x,p '&' 'not' q) is valid by QC_LANG2:def 2;
  then (p => Ex(x,q)) => 'not' All(x,'not' 'not'(p '&' 'not' q)) is valid by A1
,LUKASI_1:42;
  then (p => Ex(x,q)) => 'not' All(x,'not'(p => q)) is valid by QC_LANG2:def 2;
  hence thesis by QC_LANG2:def 5;
end;
