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 is valid iff 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 is valid iff 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 is valid iff All (x,(p => q)) is valid ) )
assume not x in still_not-bound_in q ; :: thesis: ( (Ex (x,p)) => q is valid iff All (x,(p => q)) is valid )
then ((Ex (x,p)) => q) <=> (All (x,(p => q))) is valid by Th81;
hence ( (Ex (x,p)) => q is valid iff All (x,(p => q)) is valid ) by Lm15; :: thesis: verum