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 that
A1: p in CnPos X and
A2: 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 that
A3: T is Hilbert_theory and
A4: X c= T ; :: thesis: q in T
A5: p => q in T by A2, A3, A4, Def11;
p in T by A1, A3, A4, Def11;
hence q in T by A3, A5; :: thesis: verum

end;

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