let p, q be Element of CQC-WFF ; :: thesis: for x being bound_QC-variable holds
( (Ex x,(p 'or' q)) => ((Ex x,p) 'or' (Ex x,q)) is valid & ((Ex x,p) 'or' (Ex x,q)) => (Ex x,(p 'or' q)) is valid )
let x be bound_QC-variable; :: thesis: ( (Ex x,(p 'or' q)) => ((Ex x,p) 'or' (Ex x,q)) is valid & ((Ex x,p) 'or' (Ex x,q)) => (Ex x,(p 'or' q)) is valid )
( not x in still_not-bound_in (Ex x,p) & not x in still_not-bound_in (Ex x,q) ) by Th6;
then A1: not x in still_not-bound_in ((Ex x,p) 'or' (Ex x,q)) by Th10;
( p => (Ex x,p) is valid & q => (Ex x,q) is valid ) by Th18;
then (p 'or' q) => ((Ex x,p) 'or' (Ex x,q)) is valid by Lm4;
hence (Ex x,(p 'or' q)) => ((Ex x,p) 'or' (Ex x,q)) is valid by A1, Th22; :: thesis: ((Ex x,p) 'or' (Ex x,q)) => (Ex x,(p 'or' q)) is valid
( All x,(q => (p 'or' q)) is valid & (All x,(q => (p 'or' q))) => ((Ex x,q) => (Ex x,(p 'or' q))) is valid ) by Lm6, Th26, Th38;
then A2: (Ex x,q) => (Ex x,(p 'or' q)) is valid by CQC_THE1:104;
( All x,(p => (p 'or' q)) is valid & (All x,(p => (p 'or' q))) => ((Ex x,p) => (Ex x,(p 'or' q))) is valid ) by Lm6, Th26, Th38;
then (Ex x,p) => (Ex x,(p 'or' q)) is valid by CQC_THE1:104;
hence ((Ex x,p) 'or' (Ex x,q)) => (Ex x,(p 'or' q)) is valid by A2, Lm7; :: thesis: verum