let p be Element of CQC-WFF ; :: thesis: for x being bound_QC-variable st p is valid holds
All (x,p) is valid
let x be bound_QC-variable; :: thesis: ( p is valid implies All (x,p) is valid )
A1: p => (((All (x,p)) => (All (x,p))) => p) is valid ;
not x in still_not-bound_in (All (x,p)) by Th5;
then A2: not x in still_not-bound_in ((All (x,p)) => (All (x,p))) by Th7;
assume p is valid ; :: thesis: All (x,p) is valid
then ((All (x,p)) => (All (x,p))) => p is valid by A1, CQC_THE1:65;
then ((All (x,p)) => (All (x,p))) => (All (x,p)) is valid by A2, CQC_THE1:67;
hence All (x,p) is valid by CQC_THE1:65; :: thesis: verum