Description: Axiom ax-c152220 was the original version of ax-121854, before it was
discovered (in Jan. 2007) that the shorter ax-121854 could replace it. It
appears as Axiom scheme C15' in [Megill]
p. 448 (p. 16 of the preprint).
It is based on Lemma 16 of [Tarski] p. 70
and Axiom C8 of [Monk2] p. 105,
from which it can be proved by cases. To understand this theorem more
easily, think of "-.A.xx=->..." as
informally meaning "if
and are distinct variables then..." The
antecedent becomes
false if the same variable is substituted for and , ensuring
the theorem is sound whenever this is the case. In some later theorems,
we call an antecedent of the form a "distinctor."

Interestingly, if the wff expression substituted for contains no
wff variables, the resulting statement can be proved without
invoking
this axiom. This means that even though this axiom is
metalogically
independent from the others, it is not logically independent.
Specifically, we can prove any wff-variable-free instance of axiom
ax-c152220 (from which the ax-121854 instance follows by theorem ax122234.)
The proof is by induction on formula length, using ax12eq2271 and ax12el2272
for the basis steps and ax12indn2273, ax12indi2274, and ax12inda2278 for the
induction steps. (This paragraph is true provided we use ax-c112218 in
place of ax-c11n2219.)

This axiom is obsolete and should no longer be used. It is proved above
as theorem axc152085, which should be used instead. (Contributed
by NM,
14-May-1993.) (New usage is discouraged.)