**Description: **Axiom ax-c15 33651 was the original version of ax-12 2044, before it was
discovered (in Jan. 2007) that the shorter ax-12 2044 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 33651 (from which the ax-12 2044 instance follows by theorem ax12 2303.)
The proof is by induction on formula length, using ax12eq 33703 and ax12el 33704
for the basis steps and ax12indn 33705, ax12indi 33706, and ax12inda 33710 for the
induction steps. (This paragraph is true provided we use ax-c11 33649 in
place of ax-c11n 33650.)
This axiom is obsolete and should no longer be used. It is proved above
as theorem axc15 2302, which should be used instead. (Contributed
by NM,
14-May-1993.) (New usage is discouraged.) |