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

proof

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

end;

hence a in CnCPC X by Def20; :: thesis: verum