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 ) )
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_THE1:105;
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;
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_LANG3:11;
then not x in still_not-bound_in ((All x,(p 'or' q)) '&' ('not' p)) by A1, Th9;
then ((All x,(p 'or' q)) '&' ('not' p)) => (All x,q) is valid by A2, CQC_THE1:106;
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_THE1:106;
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 Th43;
hence (p 'or' (All x,q)) => (All x,(p 'or' q)) is valid by A5, LUKASI_1:43; :: 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_1:43; :: thesis: verum