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

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

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