let p, q be Element of CQC-WFF ; :: thesis: for h being QC-formula
for x, y being bound_QC-variable st p = h . x & q = h . y & not x in still_not-bound_in h holds
(All x,p) => q is valid
let h be QC-formula; :: thesis: for x, y being bound_QC-variable st p = h . x & q = h . y & not x in still_not-bound_in h holds
(All x,p) => q is valid
let x, y be bound_QC-variable; :: thesis: ( p = h . x & q = h . y & not x in still_not-bound_in h implies (All x,p) => q is valid )
assume that
A1: p = h . x and
A2: q = h . y and
A3: not x in still_not-bound_in h ; :: thesis: (All x,p) => q is valid
A4: ((All x,p) => h) . y = ((All x,p) . y) => q by A2, Th14
.= (All x,p) => q by CQC_LANG:40 ;
not x in still_not-bound_in (All x,p) by Th5;
then A5: ( (All x,p) => p is valid & not x in still_not-bound_in ((All x,p) => h) ) by A3, Th7, CQC_THE1:105;
((All x,p) => h) . x = ((All x,p) . x) => p by A1, Th14
.= (All x,p) => p by CQC_LANG:40 ;
hence (All x,p) => q is valid by A4, A5, CQC_THE1:107; :: thesis: verum