reserve p,p1,p2,q,r,F,G,G1,G2,H,H1,H2 for ZF-formula,
  x,x1,x2,y,y1,y2,z,z1,z2,s,t for Variable,
  a,X for set;
reserve M for non empty set,
  m,m9 for Element of M,
  v,v9 for Function of VAR,M;
reserve i,j for Element of NAT;

theorem
  H is conditional implies (M,v |= H iff (M,v |= the_antecedent_of H
  implies M,v |= the_consequent_of H))
proof
  assume H is conditional;
  then H = (the_antecedent_of H) => (the_consequent_of H) by ZF_LANG:47;
  hence thesis by ZF_MODEL:18;
end;
