let p, q be Element of CQC-WFF ; :: thesis: for x being bound_QC-variable 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; :: 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 ) )
assume A1: 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 )
p => p is valid by LUKASI_1:44;
then p => (All x,p) is valid by A1, CQC_THE1:106;
then ( ((All x,p) 'or' (All x,q)) => (All x,(p 'or' q)) is valid & (p 'or' (All x,q)) => ((All x,p) 'or' (All x,q)) is valid ) by Lm10, Th43;
hence (p 'or' (All x,q)) => (All x,(p 'or' q)) is valid by LUKASI_1:43; :: thesis: (All x,(p 'or' q)) => (p 'or' (All x,q)) is valid
( (All x,(p 'or' q)) => (p 'or' q) is valid & (p 'or' q) => (('not' p) => q) is valid ) by Lm11, CQC_THE1:105;
then (All x,(p 'or' q)) => (('not' p) => q) is valid by LUKASI_1:43;
then A2: ((All x,(p 'or' q)) '&' ('not' p)) => q is valid by Th1;
( not x in still_not-bound_in ('not' p) & not x in still_not-bound_in (All x,(p 'or' q)) ) by A1, Th5, QC_LANG3:11;
then not x in still_not-bound_in ((All x,(p 'or' q)) '&' ('not' p)) by Th9;
then ((All x,(p 'or' q)) '&' ('not' p)) => (All x,q) is valid by A2, CQC_THE1:106;
then ( (All x,(p 'or' q)) => (('not' p) => (All x,q)) is valid & (('not' p) => (All x,q)) => (p 'or' (All x,q)) is valid ) by Lm12, Th3;
hence (All x,(p 'or' q)) => (p 'or' (All x,q)) is valid by LUKASI_1:43; :: thesis: verum