Hi, I recently uploaded a couple of theorems to my Mathbox, starting with http://us2.metamath.org:88/mpeuni/wl-section-nf.htm, that demonstrate how things could have developed, if one had picked http://us2.metamath.org:88/mpeuni/nf2.html instead of http://us2.metamath.org:88/mpeuni/df-nf.html as the defining term for 'Not Free (Ⅎ)'. The results show, that one can expect in predicate logic an overall reduction of the usage of http://us2.metamath.org:88/mpeuni/ax-10.html, and perhaps a marginal decrease in use of http://us2.metamath.org:88/mpeuni/ax-12.html. This certainly positive result is due to the more symmetric structure of nf2 wrt to negation, and the avoidance of mixed quantified and not quantified variables. But it comes with a price, too. (a) There is more friction in the transition between hb* and nf* style theorems; (b) Axioms like ax-5 need the old style for full exploitation, so there is a sort of disruption in technique present. You will find more details in a http://us2.metamath.org:88/mpeuni/wl-section-nf.html. But these are my thoughts. Let me hear how you think about a change of definition of 'Not Free'. Wolf Lammen
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