let A be QC-alphabet ; :: thesis: for p being QC-formula of A st p is conditional holds
still_not-bound_in p = (still_not-bound_in (the_antecedent_of p)) \/ (still_not-bound_in (the_consequent_of p))
let p be QC-formula of A; :: thesis: ( p is conditional implies still_not-bound_in p = (still_not-bound_in (the_antecedent_of p)) \/ (still_not-bound_in (the_consequent_of p)) )
set p1 = the_antecedent_of p;
set p2 = the_consequent_of p;
assume p is conditional ; :: thesis: still_not-bound_in p = (still_not-bound_in (the_antecedent_of p)) \/ (still_not-bound_in (the_consequent_of p))
then p = (the_antecedent_of p) => (the_consequent_of p) by QC_LANG2:38;
then p = 'not' ((the_antecedent_of p) '&' ('not' (the_consequent_of p))) by QC_LANG2:def 2;
then still_not-bound_in p = still_not-bound_in ((the_antecedent_of p) '&' ('not' (the_consequent_of p))) by Th7
.= (still_not-bound_in (the_antecedent_of p)) \/ (still_not-bound_in ('not' (the_consequent_of p))) by Th10
.= (still_not-bound_in (the_antecedent_of p)) \/ (still_not-bound_in (the_consequent_of p)) by Th7 ;
hence still_not-bound_in p = (still_not-bound_in (the_antecedent_of p)) \/ (still_not-bound_in (the_consequent_of p)) ; :: thesis: verum