let p, q be Element of CQC-WFF ; :: thesis: for x being bound_QC-variable holds
( (All x,(p '&' q)) => ((All x,p) '&' (All x,q)) is valid & ((All x,p) '&' (All x,q)) => (All x,(p '&' q)) is valid )
let x be bound_QC-variable; :: thesis: ( (All x,(p '&' q)) => ((All x,p) '&' (All x,q)) is valid & ((All x,p) '&' (All x,q)) => (All x,(p '&' q)) is valid )
( (All x,p) => p is valid & (All x,q) => q is valid ) by CQC_THE1:105;
then A1: ((All x,p) '&' (All x,q)) => (p '&' q) is valid by Lm5;
( All x,((p '&' q) => q) is valid & (All x,((p '&' q) => q)) => ((All x,(p '&' q)) => (All x,q)) is valid ) by Lm1, Th26, Th34;
then A2: (All x,(p '&' q)) => (All x,q) is valid by CQC_THE1:104;
( All x,((p '&' q) => p) is valid & (All x,((p '&' q) => p)) => ((All x,(p '&' q)) => (All x,p)) is valid ) by Lm1, Th26, Th34;
then (All x,(p '&' q)) => (All x,p) is valid by CQC_THE1:104;
hence (All x,(p '&' q)) => ((All x,p) '&' (All x,q)) is valid by A2, Lm3; :: thesis: ((All x,p) '&' (All x,q)) => (All x,(p '&' q)) is valid
( not x in still_not-bound_in (All x,p) & not x in still_not-bound_in (All x,q) ) by Th5;
then not x in still_not-bound_in ((All x,p) '&' (All x,q)) by Th9;
hence ((All x,p) '&' (All x,q)) => (All x,(p '&' q)) is valid by A1, CQC_THE1:106; :: thesis: verum