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))) <=> ((All (x,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))) <=> ((All (x,p)) '&' (All (x,q))) is valid
let x be bound_QC-variable of A; :: thesis: (All (x,(p '&' q))) <=> ((All (x,p)) '&' (All (x,q))) is valid
( (All (x,(p '&' q))) => ((All (x,p)) '&' (All (x,q))) is valid & ((All (x,p)) '&' (All (x,q))) => (All (x,(p '&' q))) is valid ) by Th36;
hence (All (x,(p '&' q))) <=> ((All (x,p)) '&' (All (x,q))) is valid by Lm14; :: thesis: verum