let Al be QC-alphabet ; :: thesis: for X being Subset of (CQC-WFF Al)
for p, q being Element of CQC-WFF Al
for x being bound_QC-variable of Al 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 Subset of (CQC-WFF Al); :: thesis: for p, q being Element of CQC-WFF Al
for x being bound_QC-variable of Al 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 Al; :: thesis: for x being bound_QC-variable of Al 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 of Al; :: 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
for T being Subset of (CQC-WFF Al) st T is being_a_theory & X c= T holds
p => (All (x,q)) in T

proof

let T be Subset of (CQC-WFF Al); :: thesis: ( T is being_a_theory & X c= T implies p => (All (x,q)) in T )
assume that
A3: T is being_a_theory and
A4: X c= T ; :: thesis: p => (All (x,q)) in T
p => q in T by A1, A3, A4, Def2;
hence p => (All (x,q)) in T by A2, A3; :: thesis: verum

end;

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