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)) 'or' (All (x,q))) => (All (x,(p 'or' 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)) 'or' (All (x,q))) => (All (x,(p 'or' q))) is valid
let x be bound_QC-variable of A; :: thesis: ((All (x,p)) 'or' (All (x,q))) => (All (x,(p 'or' q))) is valid
( All (x,(q => (p 'or' q))) is valid & (All (x,(q => (p 'or' q)))) => ((All (x,q)) => (All (x,(p 'or' q)))) is valid ) by Lm6, Th23, Th30;
then A1: (All (x,q)) => (All (x,(p 'or' q))) is valid by CQC_THE1:65;
( All (x,(p => (p 'or' q))) is valid & (All (x,(p => (p 'or' q)))) => ((All (x,p)) => (All (x,(p 'or' q)))) is valid ) by Lm6, Th23, Th30;
then (All (x,p)) => (All (x,(p 'or' q))) is valid by CQC_THE1:65;
hence ((All (x,p)) 'or' (All (x,q))) => (All (x,(p 'or' q))) is valid by A1, Lm7; :: thesis: verum