| Description: Axiom ax-c15 37164 was the original version of ax-12 2170, before it was
discovered (in Jan. 2007) that the shorter ax-12 2170 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 37164 (from which the ax-12 2170 instance follows by Theorem ax12 2421.)
The proof is by induction on formula length, using ax12eq 37216 and ax12el 37217
for the basis steps and ax12indn 37218, ax12indi 37219, and ax12inda 37223 for the
induction steps. (This paragraph is true provided we use ax-c11 37162 in
place of ax-c11n 37163.)
This axiom is obsolete and should no longer be used. It is proved above
as Theorem axc15 2420, which should be used instead. (Contributed
by NM,
14-May-1993.) (New usage is discouraged.) |
