let A be QC-alphabet ; :: thesis: for p being QC-formula of A st p is biconditional holds
still_not-bound_in p = (still_not-bound_in (the_left_side_of p)) \/ (still_not-bound_in (the_right_side_of p))
let p be QC-formula of A; :: thesis: ( p is biconditional implies still_not-bound_in p = (still_not-bound_in (the_left_side_of p)) \/ (still_not-bound_in (the_right_side_of p)) )
set p1 = the_left_side_of p;
set p2 = the_right_side_of p;
assume p is biconditional ; :: thesis: still_not-bound_in p = (still_not-bound_in (the_left_side_of p)) \/ (still_not-bound_in (the_right_side_of p))
then p = (the_left_side_of p) <=> (the_right_side_of p) by QC_LANG2:39;
then p = ((the_left_side_of p) => (the_right_side_of p)) '&' ((the_right_side_of p) => (the_left_side_of p)) by QC_LANG2:def 4;
then still_not-bound_in p = (still_not-bound_in ((the_left_side_of p) => (the_right_side_of p))) \/ (still_not-bound_in ((the_right_side_of p) => (the_left_side_of p))) by Th10
.= ((still_not-bound_in (the_left_side_of p)) \/ (still_not-bound_in (the_right_side_of p))) \/ (still_not-bound_in ((the_right_side_of p) => (the_left_side_of p))) by Th16
.= ((still_not-bound_in (the_left_side_of p)) \/ (still_not-bound_in (the_right_side_of p))) \/ ((still_not-bound_in (the_left_side_of p)) \/ (still_not-bound_in (the_right_side_of p))) by Th16
.= (still_not-bound_in (the_left_side_of p)) \/ (still_not-bound_in (the_right_side_of p)) ;
hence still_not-bound_in p = (still_not-bound_in (the_left_side_of p)) \/ (still_not-bound_in (the_right_side_of p)) ; :: thesis: verum