let X, Y be Subset of MC-wff; :: thesis: ( X c= Y implies CnIPC X c= CnIPC Y )
assume A1: X c= Y ; :: thesis: CnIPC X c= CnIPC Y
thus CnIPC X c= CnIPC Y :: thesis: verum

proof

let a be object ; :: according to TARSKI:def 3 :: thesis: ( not a in CnIPC X or a in CnIPC Y )
assume A2: a in CnIPC X ; :: thesis: a in CnIPC Y
then reconsider t = a as Element of MC-wff ;
for T being Subset of MC-wff st T is IPC_theory & Y c= T holds
t in T

proof

let T be Subset of MC-wff; :: thesis: ( T is IPC_theory & Y c= T implies t in T )
assume that
A3: T is IPC_theory and
A4: Y c= T ; :: thesis: t in T
X c= T by A1, A4;
hence t in T by A2, A3, Def15; :: thesis: verum

end;

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

end;