let p, q be Element of CQC-WFF ; :: thesis: for x being bound_QC-variable holds (All x,(p => q)) => ((Ex x,p) => (Ex x,q)) is valid
let x be bound_QC-variable; :: thesis: (All x,(p => q)) => ((Ex x,p) => (Ex 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 Th2;
q => (Ex x,q) is valid by Th18;
then (p '&' (All x,(p => q))) => (Ex x,q) is valid by A1, LUKASI_1:43;
then A2: p => ((All x,(p => q)) => (Ex x,q)) is valid by Th3;
( not x in still_not-bound_in (All x,(p => q)) & not x in still_not-bound_in (Ex x,q) ) by Th5, Th6;
then not x in still_not-bound_in ((All x,(p => q)) => (Ex x,q)) by Th7;
then (Ex x,p) => ((All x,(p => q)) => (Ex x,q)) is valid by A2, Th22;
hence (All x,(p => q)) => ((Ex x,p) => (Ex x,q)) is valid by LUKASI_1:48; :: thesis: verum