let X be Subset of MC-wff; :: thesis: CnIPC (CnIPC X) c= CnIPC X
let a be object ; :: according to TARSKI:def 3 :: thesis: ( not a in CnIPC (CnIPC X) or a in CnIPC X )
assume A1: a in CnIPC (CnIPC X) ; :: thesis: a in CnIPC X
then reconsider t = a as Element of MC-wff ;
for T being Subset of MC-wff st T is IPC_theory & X c= T holds
t in T

proof

let T be Subset of MC-wff; :: thesis: ( T is IPC_theory & X c= T implies t in T )
assume that
A2: T is IPC_theory and
A3: X c= T ; :: thesis: t in T
CnIPC X c= T by A2, A3, Th11;
hence t in T by A1, A2, Def15; :: thesis: verum

end;

hence a in CnIPC X by Def15; :: thesis: verum