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))) => ((Ex (x,p)) => (Ex (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))) => ((Ex (x,p)) => (Ex (x,q))) is valid
let x be bound_QC-variable of A; :: thesis: (All (x,(p => q))) => ((Ex (x,p)) => (Ex (x,q))) is valid
(All (x,(p => q))) => (p => q) is valid by CQC_THE1:66;
then A1: (p '&' (All (x,(p => q)))) => q is valid by Th2;
q => (Ex (x,q)) is valid by Th15;
then (p '&' (All (x,(p => q)))) => (Ex (x,q)) is valid by A1, LUKASI_1:42;
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, Th19;
hence (All (x,(p => q))) => ((Ex (x,p)) => (Ex (x,q))) is valid by LUKASI_1:44; :: thesis: verum