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 p holds
( (p 'or' (All (x,q))) => (All (x,(p 'or' q))) is valid & (All (x,(p 'or' q))) => (p 'or' (All (x,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 p holds
( (p 'or' (All (x,q))) => (All (x,(p 'or' q))) is valid & (All (x,(p 'or' q))) => (p 'or' (All (x,q))) is valid )
let x be bound_QC-variable of A; :: thesis: ( not x in still_not-bound_in p implies ( (p 'or' (All (x,q))) => (All (x,(p 'or' q))) is valid & (All (x,(p 'or' q))) => (p 'or' (All (x,q))) is valid ) )
A1: not x in still_not-bound_in (All (x,(p 'or' q))) by Th5;
( (All (x,(p 'or' q))) => (p 'or' q) is valid & (p 'or' q) => (('not' p) => q) is valid ) by Lm11, CQC_THE1:66;
then (All (x,(p 'or' q))) => (('not' p) => q) is valid by LUKASI_1:42;
then A2: ((All (x,(p 'or' q))) '&' ('not' p)) => q is valid by Th1;
assume A3: not x in still_not-bound_in p ; :: thesis: ( (p 'or' (All (x,q))) => (All (x,(p 'or' q))) is valid & (All (x,(p 'or' q))) => (p 'or' (All (x,q))) is valid )
then not x in still_not-bound_in ('not' p) by QC_LANG3:7;
then not x in still_not-bound_in ((All (x,(p 'or' q))) '&' ('not' p)) by A1, Th8;
then ((All (x,(p 'or' q))) '&' ('not' p)) => (All (x,q)) is valid by A2, CQC_THE1:67;
then A4: (All (x,(p 'or' q))) => (('not' p) => (All (x,q))) is valid by Th3;
p => p is valid ;
then p => (All (x,p)) is valid by A3, CQC_THE1:67;
then A5: (p 'or' (All (x,q))) => ((All (x,p)) 'or' (All (x,q))) is valid by Lm10;
((All (x,p)) 'or' (All (x,q))) => (All (x,(p 'or' q))) is valid by Th39;
hence (p 'or' (All (x,q))) => (All (x,(p 'or' q))) is valid by A5, LUKASI_1:42; :: thesis: (All (x,(p 'or' q))) => (p 'or' (All (x,q))) is valid
(('not' p) => (All (x,q))) => (p 'or' (All (x,q))) is valid by Lm12;
hence (All (x,(p 'or' q))) => (p 'or' (All (x,q))) is valid by A4, LUKASI_1:42; :: thesis: verum