|Description: Axiom of Substitution.
One of the 5 equality axioms of predicate
calculus. The final consequent ∀x(x = y →
φ) is a way of
substituted for x in wff
φ " (cf. sb6 2099).
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-11o 2141 ("o" for "old") and was
replaced with this shorter ax-11 1746 in Jan. 2007. The old axiom is
from this one as theorem ax11o 1994. Conversely, this axiom is proved from
ax-11o 2141 as theorem ax11 2155.
Juha Arpiainen proved the metalogical independence of this axiom (in the
form of the older axiom ax-11o 2141) from the others on 19-Jan-2006. See
item 9a at http://us.metamath.org/award2003.html.
See ax11v 2096 and ax11v2 1992 for other equivalents of this axiom that
this axiom) have distinct variable restrictions.
This axiom scheme is logically redundant (see ax11w 1721) but is used as an
auxiliary axiom to achieve metalogical completeness. (Contributed by NM,