**Description: **Axiom ax-11o 2141 ("o" for "old") was the
original version of ax-11 1746,
before it was discovered (in Jan. 2007) that the shorter ax-11 1746 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 "¬ ∀*x**x* = *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 ¬ ∀*x**x* = *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-11o 2141 (from which the ax-11 1746 instance follows by theorem ax11 2155.)
The proof is by induction on formula length, using ax11eq 2193 and ax11el 2194
for the basis steps and ax11indn 2195, ax11indi 2196, and ax11inda 2200 for the
induction steps. (This paragraph is true provided we use ax-10o 2139 in
place of ax-10 2140.)
This axiom is obsolete and should no longer be used. It is proved above
as theorem ax11o 1994. (Contributed by NM, 5-Aug-1993.)
(New usage is discouraged.) |