let X be Subset of MC-wff; :: thesis: for p, q being Element of MC-wff st p in CnS4 X & p => q in CnS4 X holds
q in CnS4 X
let p, q be Element of MC-wff ; :: thesis: ( p in CnS4 X & p => q in CnS4 X implies q in CnS4 X )
assume that
A1: p in CnS4 X and
A2: p => q in CnS4 X ; :: thesis: q in CnS4 X
for T being Subset of MC-wff st T is S4_theory & X c= T holds
q in T

proof

let T be Subset of MC-wff; :: thesis: ( T is S4_theory & X c= T implies q in T )
assume that
A3: T is S4_theory and
A4: X c= T ; :: thesis: q in T
A5: p => q in T by A2, A3, A4, Def23;
p in T by A1, A3, A4, Def23;
hence q in T by A3, A5; :: thesis: verum

end;

hence q in CnS4 X by Def23; :: thesis: verum