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 q holds
((Ex (x,p)) => q) <=> (All (x,(p => 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 q holds
((Ex (x,p)) => q) <=> (All (x,(p => q))) is valid
let x be bound_QC-variable of A; :: thesis: ( not x in still_not-bound_in q implies ((Ex (x,p)) => q) <=> (All (x,(p => q))) is valid )
assume not x in still_not-bound_in q ; :: thesis: ((Ex (x,p)) => q) <=> (All (x,(p => q))) is valid
then ( ((Ex (x,p)) => q) => (All (x,(p => q))) is valid & (All (x,(p => q))) => ((Ex (x,p)) => q) is valid ) by Th80;
hence ((Ex (x,p)) => q) <=> (All (x,(p => q))) is valid by Lm14; :: thesis: verum