let A be QC-alphabet ; :: thesis: for p, q being Element of CQC-WFF A
for x being bound_QC-variable of A 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 p, q be Element of CQC-WFF A; :: thesis: for x being bound_QC-variable of A 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 of A; :: 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 Th9;
( p => (Ex (x,p)) is valid & q => (Ex (x,q)) is valid ) by Th15;
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, Th19; :: 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, Th23, Th34;
then A2: (Ex (x,q)) => (Ex (x,(p 'or' q))) is valid by CQC_THE1:65;
( 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, Th23, Th34;
then (Ex (x,p)) => (Ex (x,(p 'or' q))) is valid by CQC_THE1:65;
hence ((Ex (x,p)) 'or' (Ex (x,q))) => (Ex (x,(p 'or' q))) is valid by A2, Lm7; :: thesis: verum