let A be QC-alphabet ; :: thesis: for p, q being Element of CQC-WFF A
for x being bound_QC-variable of A st not x in still_not-bound_in q holds
( ((Ex (x,p)) => q) => (All (x,(p => q))) is valid & (All (x,(p => q))) => ((Ex (x,p)) => q) is valid )
let p, q be Element of CQC-WFF A; :: thesis: for x being bound_QC-variable of A st not x in still_not-bound_in q holds
( ((Ex (x,p)) => q) => (All (x,(p => q))) is valid & (All (x,(p => q))) => ((Ex (x,p)) => q) is valid )
let x be bound_QC-variable of A; :: thesis: ( not x in still_not-bound_in q implies ( ((Ex (x,p)) => q) => (All (x,(p => q))) is valid & (All (x,(p => q))) => ((Ex (x,p)) => q) is valid ) )
assume A1: not x in still_not-bound_in q ; :: thesis: ( ((Ex (x,p)) => q) => (All (x,(p => q))) is valid & (All (x,(p => q))) => ((Ex (x,p)) => q) is valid )
p => (Ex (x,p)) is valid by Th15;
then A2: ((Ex (x,p)) => q) => (p => q) is valid by LUKASI_1:41;
not x in still_not-bound_in (Ex (x,p)) by Th6;
then not x in still_not-bound_in ((Ex (x,p)) => q) by A1, Th7;
hence ((Ex (x,p)) => q) => (All (x,(p => q))) is valid by A2, CQC_THE1:67; :: thesis: (All (x,(p => q))) => ((Ex (x,p)) => q) is valid
(All (x,(p => q))) => ((Ex (x,p)) => (Ex (x,q))) is valid by Th34;
then A3: ((All (x,(p => q))) '&' (Ex (x,p))) => (Ex (x,q)) is valid by Th1;
(Ex (x,q)) => q is valid by A1, Th20;
then ((All (x,(p => q))) '&' (Ex (x,p))) => q is valid by A3, LUKASI_1:42;
hence (All (x,(p => q))) => ((Ex (x,p)) => q) is valid by Th3; :: thesis: verum