let X be Subset of HP-WFF ; :: thesis: for p, q being Element of HP-WFF st p in CnPos X & p => q in CnPos X holds
q in CnPos X
let p, q be Element of HP-WFF ; :: thesis: ( p in CnPos X & p => q in CnPos X implies q in CnPos X )
assume A1: ( p in CnPos X & p => q in CnPos X ) ; :: thesis: q in CnPos X
for T being Subset of HP-WFF st T is Hilbert_theory & X c= T holds
q in T

proof

let T be Subset of HP-WFF ; :: thesis: ( T is Hilbert_theory & X c= T implies q in T )
assume A2: ( T is Hilbert_theory & X c= T ) ; :: thesis: q in T
then ( p in T & p => q in T ) by A1, Def11;
hence q in T by A2, Def10; :: thesis: verum

end;

hence q in CnPos X by Def11; :: thesis: verum