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
let x be bound_QC-variable; :: thesis: (All x,(p => q)) => ((All x,p) => (All x,q)) is valid
(All x,(p => q)) => (p => q) is valid by CQC_THE1:105;
then A1: p => ((All x,(p => q)) => q) is valid by LUKASI_1:48;
(All x,p) => p is valid by CQC_THE1:105;
then (All x,p) => ((All x,(p => q)) => q) is valid by A1, LUKASI_1:43;
then A2: ((All x,(p => q)) '&' (All x,p)) => q is valid by Th2;
( not x in still_not-bound_in (All x,(p => q)) & not x in still_not-bound_in (All x,p) ) by Th5;
then not x in still_not-bound_in ((All x,(p => q)) '&' (All x,p)) by Th9;
then ((All x,(p => q)) '&' (All x,p)) => (All x,q) is valid by A2, CQC_THE1:106;
hence (All x,(p => q)) => ((All x,p) => (All x,q)) is valid by Th3; :: thesis: verum