**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 2086). 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 35892 and was replaced with this
shorter ax-12 2169 in Jan. 2007. The old axiom is proved from
this one as
theorem axc15 2440. Conversely, this axiom is proved from ax-c15 35892 as
theorem ax12 2442.
Juha Arpiainen proved the metalogical independence of this axiom (in the
form of the older axiom ax-c15 35892) from the others on 19-Jan-2006.
See
item 9a at https://us.metamath.org/award2003.html 35892.
See ax12v 2170 and ax12v2 2171 for other equivalents of this axiom that
(unlike
this axiom) have distinct variable restrictions.
This axiom scheme is logically redundant (see ax12w 2130) but is used as an
auxiliary axiom scheme to achieve scheme completeness. (Contributed by
NM, 22-Jan-2007.) (New usage is discouraged.) |