**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 2427). 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 33993 and was replaced with this
shorter ax-12 2045 in Jan. 2007. The old axiom is proved from
this one as
theorem axc15 2301. Conversely, this axiom is proved from ax-c15 33993 as
theorem ax12 2302.
Juha Arpiainen proved the metalogical independence of this axiom (in the
form of the older axiom ax-c15 33993) from the others on 19-Jan-2006.
See
item 9a at http://us.metamath.org/award2003.html.
See ax12v 2046 and ax12v2 2047 for other equivalents of this axiom that
(unlike
this axiom) have distinct variable restrictions.
This axiom scheme is logically redundant (see ax12w 2008) but is used as an
auxiliary axiom scheme to achieve scheme completeness. (Contributed by
NM, 22-Jan-2007.) (New usage is discouraged.) |