let X be Subset of MC-wff; :: thesis: for p, q being Element of MC-wff st p in CnCPC X & p => q in CnCPC X holds
q in CnCPC X
let p, q be Element of MC-wff ; :: thesis: ( p in CnCPC X & p => q in CnCPC X implies q in CnCPC X )
assume that
A1: p in CnCPC X and
A2: p => q in CnCPC X ; :: thesis: q in CnCPC X
for T being Subset of MC-wff st T is CPC_theory & X c= T holds
q in T
proof
let T be Subset of MC-wff; :: thesis: ( T is CPC_theory & X c= T implies q in T )
assume that
A3: T is CPC_theory and
A4: X c= T ; :: thesis: q in T
A5: p => q in T by A2, A3, A4, Def20;
p in T by A1, A3, A4, Def20;
hence q in T by A3, A5; :: thesis: verum
end;
hence q in CnCPC X by Def20; :: thesis: verum