let X be Subset of MC-wff; :: thesis: for p, q being Element of MC-wff st p in CnIPC X & p => q in CnIPC X holds
q in CnIPC X
let p, q be Element of MC-wff ; :: thesis: ( p in CnIPC X & p => q in CnIPC X implies q in CnIPC X )
assume that
A1: p in CnIPC X and
A2: p => q in CnIPC X ; :: thesis: q in CnIPC X
for T being Subset of MC-wff st T is IPC_theory & X c= T holds
q in T

proof

let T be Subset of MC-wff; :: thesis: ( T is IPC_theory & X c= T implies q in T )
assume that
A3: T is IPC_theory and
A4: X c= T ; :: thesis: q in T
A5: p => q in T by A2, A3, A4, Def15;
p in T by A1, A3, A4, Def15;
hence q in T by A3, A5; :: thesis: verum

end;

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