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 Th34:
  All(x,p => q) => (Ex(x,p) => Ex(x,q)) is valid
proof
  All(x,p => q) => (p => q) is valid by CQC_THE1:66;
  then
A1: (p '&' All(x,p => q)) => q is valid by Th2;
  q => Ex(x,q) is valid by Th15;
  then (p '&' All(x,p => q)) => Ex(x,q) is valid by A1,LUKASI_1:42;
  then
A2: p => (All(x,p => q) => Ex(x,q)) is valid by Th3;
  ( not x in still_not-bound_in All(x,p => q))& not x in
  still_not-bound_in Ex(x,q) by Th5,Th6;
  then not x in still_not-bound_in All(x,p => q) => Ex(x,q) by Th7;
  then Ex(x,p) => (All(x,p => q) => Ex(x,q)) is valid by A2,Th19;
  hence thesis by LUKASI_1:44;
end;
