let A be QC-alphabet ; for p being Element of CQC-WFF A
for x, y being bound_QC-variable of A holds
( (All (x,(All (y,p)))) => (All (y,(All (x,p)))) is valid & (All (x,y,p)) => (All (y,x,p)) is valid )
let p be Element of CQC-WFF A; for x, y being bound_QC-variable of A holds
( (All (x,(All (y,p)))) => (All (y,(All (x,p)))) is valid & (All (x,y,p)) => (All (y,x,p)) is valid )
let x, y be bound_QC-variable of A; ( (All (x,(All (y,p)))) => (All (y,(All (x,p)))) is valid & (All (x,y,p)) => (All (y,x,p)) is valid )
not y in still_not-bound_in (All (y,p))
by Th5;
then A1:
not y in still_not-bound_in (All (x,(All (y,p))))
by Th5;
( All (x,((All (y,p)) => p)) is valid & (All (x,((All (y,p)) => p))) => ((All (x,(All (y,p)))) => (All (x,p))) is valid )
by Th23, Th30, CQC_THE1:66;
then
(All (x,(All (y,p)))) => (All (x,p)) is valid
by CQC_THE1:65;
hence
(All (x,(All (y,p)))) => (All (y,(All (x,p)))) is valid
by A1, CQC_THE1:67; (All (x,y,p)) => (All (y,x,p)) is valid
then
(All (x,y,p)) => (All (y,(All (x,p)))) is valid
by QC_LANG2:14;
hence
(All (x,y,p)) => (All (y,x,p)) is valid
by QC_LANG2:14; verum