let A be QC-alphabet ; :: thesis: for p, q being Element of CQC-WFF A
for x being bound_QC-variable of A holds (All (x,(p '&' q))) => (p '&' (All (x,q))) is valid
let p, q be Element of CQC-WFF A; :: thesis: for x being bound_QC-variable of A holds (All (x,(p '&' q))) => (p '&' (All (x,q))) is valid
let x be bound_QC-variable of A; :: thesis: (All (x,(p '&' q))) => (p '&' (All (x,q))) is valid
A1: (All (x,(p '&' q))) => (p '&' q) is valid by CQC_THE1:66;
A2: not x in still_not-bound_in (All (x,(p '&' q))) by Th5;
(p '&' q) => q is valid by Lm1;
then (All (x,(p '&' q))) => q is valid by A1, LUKASI_1:42;
then A3: (All (x,(p '&' q))) => (All (x,q)) is valid by A2, CQC_THE1:67;
(p '&' q) => p is valid by Lm1;
then (All (x,(p '&' q))) => p is valid by A1, LUKASI_1:42;
hence (All (x,(p '&' q))) => (p '&' (All (x,q))) is valid by A3, Lm3; :: thesis: verum