|Description: Axiom of Substitution.
One of the 5 equality axioms of predicate
calculus. The final consequent
is a way of
substituted for in wff
" (cf. sb6 2152).
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 2195 ("o" for "old") and was
replaced with this shorter ax-11 1757 in Jan. 2007. The old axiom is
from this one as theorem ax11o 2051. Conversely, this axiom is proved from
ax-11o 2195 as theorem ax11 2209.
Juha Arpiainen proved the metalogical independence of this axiom (in the
form of the older axiom ax-11o 2195) from the others on 19-Jan-2006. See
item 9a at http://us.metamath.org/award2003.html.
See ax11v 2149 and ax11v2 2048 for other equivalents of this axiom that
this axiom) have distinct variable restrictions.
This axiom scheme is logically redundant (see ax11w 1732) but is used as an
auxiliary axiom to achieve metalogical completeness. (Contributed by NM,