| Description: Axiom ax-c15 34174 was the original version of ax-12 2047, before it was
discovered (in Jan. 2007) that the shorter ax-12 2047 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 "¬ ∀𝑥𝑥 = 𝑦 →..." as informally meaning
"if
𝑥 and 𝑦 are distinct variables
then..." The antecedent becomes
false if the same variable is substituted for 𝑥 and 𝑦,
ensuring
the theorem is sound whenever this is the case. In some later theorems,
we call an antecedent of the form ¬ ∀𝑥𝑥 = 𝑦 a "distinctor."
Interestingly, if the wff expression substituted for 𝜑 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-c15 34174 (from which the ax-12 2047 instance follows by theorem ax12 2304.)
The proof is by induction on formula length, using ax12eq 34226 and ax12el 34227
for the basis steps and ax12indn 34228, ax12indi 34229, and ax12inda 34233 for the
induction steps. (This paragraph is true provided we use ax-c11 34172 in
place of ax-c11n 34173.)
This axiom is obsolete and should no longer be used. It is proved above
as theorem axc15 2303, which should be used instead. (Contributed
by NM,
14-May-1993.) (New usage is discouraged.) |
