**Description: **Axiom of Substitution.
One of the 5 equality axioms of predicate
calculus. The final consequent ∀𝑥(𝑥 = 𝑦 → 𝜑) is a way of
expressing "𝑦 substituted for 𝑥 in wff
𝜑
" (cf. sb6 2312). 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.
The original version of this axiom was ax-c15 32694 and was replaced with this
shorter ax-12 1983 in Jan. 2007. The old axiom is proved from
this one as
theorem axc15 2187. Conversely, this axiom is proved from ax-c15 32694 as
theorem ax12 2224.
Juha Arpiainen proved the metalogical independence of this axiom (in the
form of the older axiom ax-c15 32694) from the others on 19-Jan-2006.
See
item 9a at http://us.metamath.org/award2003.html.
See ax12v 1984 and ax12v2 1985 for other equivalents of this axiom that
(unlike
this axiom) have distinct variable restrictions.
This axiom scheme is logically redundant (see ax12w 1957) but is used as an
auxiliary axiom scheme to achieve metalogical completeness. (Contributed
by NM, 22-Jan-2007.) (New usage is
discouraged.) |