let A be QC-alphabet ; :: thesis: for p, q being Element of CQC-WFF A
for h being QC-formula of A
for x, y being bound_QC-variable of A st p = h . x & q = h . y & not x in still_not-bound_in h holds
(All (x,p)) => q is valid
let p, q be Element of CQC-WFF A; :: thesis: for h being QC-formula of A
for x, y being bound_QC-variable of A 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 of A; :: thesis: for x, y being bound_QC-variable of A 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 of A; :: 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, Th12
.= (All (x,p)) => q by CQC_LANG:27 ;
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_THE1:66;
((All (x,p)) => h) . x = ((All (x,p)) . x) => p by A1, Th12
.= (All (x,p)) => p by CQC_LANG:27 ;
hence (All (x,p)) => q is valid by A4, A5, CQC_THE1:68; :: thesis: verum