let A be QC-alphabet ; :: thesis: for p, q being Element of CQC-WFF A
for x being bound_QC-variable of A st not x in still_not-bound_in p holds
(p '&' (All (x,q))) => (All (x,(p '&' q))) is valid
let p, q be Element of CQC-WFF A; :: thesis: for x being bound_QC-variable of A st not x in still_not-bound_in p holds
(p '&' (All (x,q))) => (All (x,(p '&' q))) is valid
let x be bound_QC-variable of A; :: thesis: ( not x in still_not-bound_in p implies (p '&' (All (x,q))) => (All (x,(p '&' q))) is valid )
assume A1: not x in still_not-bound_in p ; :: thesis: (p '&' (All (x,q))) => (All (x,(p '&' q))) is valid
(All (x,q)) => q is valid by CQC_THE1:66;
then A2: (p '&' (All (x,q))) => (p '&' q) is valid by Lm9;
not x in still_not-bound_in (All (x,q)) by Th5;
then not x in still_not-bound_in (p '&' (All (x,q))) by A1, Th8;
hence (p '&' (All (x,q))) => (All (x,(p '&' q))) is valid by A2, CQC_THE1:67; :: thesis: verum