let A be QC-alphabet ; :: thesis: for p, q being Element of CQC-WFF A
for x being bound_QC-variable of A st p => q is valid & not x in still_not-bound_in q holds
(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 p => q is valid & not x in still_not-bound_in q holds
(Ex (x,p)) => q is valid
let x be bound_QC-variable of A; :: thesis: ( p => q is valid & not x in still_not-bound_in q implies (Ex (x,p)) => q is valid )
assume ( p => q is valid & not x in still_not-bound_in q ) ; :: thesis: (Ex (x,p)) => q is valid
then ( ('not' q) => ('not' p) is valid & not x in still_not-bound_in ('not' q) ) by LUKASI_1:52, QC_LANG3:7;
then ('not' q) => (All (x,('not' p))) is valid by CQC_THE1:67;
then ('not' (All (x,('not' p)))) => ('not' ('not' q)) is valid by LUKASI_1:52;
then (Ex (x,p)) => ('not' ('not' q)) is valid by QC_LANG2:def 5;
hence (Ex (x,p)) => q is valid by LUKASI_1:55; :: thesis: verum