let X be Subset of CQC-WFF ; :: thesis: for p, q being Element of CQC-WFF
for x being bound_QC-variable st p => q in Cn X & not x in still_not-bound_in p holds
p => (All x,q) in Cn X
let p, q be Element of CQC-WFF ; :: thesis: for x being bound_QC-variable st p => q in Cn X & not x in still_not-bound_in p holds
p => (All x,q) in Cn X
let x be bound_QC-variable; :: thesis: ( p => q in Cn X & not x in still_not-bound_in p implies p => (All x,q) in Cn X )
assume that
A1: p => q in Cn X and
A2: not x in still_not-bound_in p ; :: thesis: p => (All x,q) in Cn X
A3: for T being Subset of CQC-WFF st T is being_a_theory & X c= T holds
p => (All x,q) in T

proof

let T be Subset of CQC-WFF ; :: thesis: ( T is being_a_theory & X c= T implies p => (All x,q) in T )
assume that
A4: T is being_a_theory and
A5: X c= T ; :: thesis: p => (All x,q) in T
A6: p => q in T by A1, A4, A5, Def2;
thus p => (All x,q) in T by A2, A4, A6, Def1; :: thesis: verum

end;

thus p => (All x,q) in Cn X by A3, Def2; :: thesis: verum