**Description: **Axiom of Existence. One
of the equality and substitution axioms of
predicate calculus with equality. This axiom tells us is that at least
one thing exists. In this form (not requiring that 𝑥 and 𝑦 be
distinct) it was used in an axiom system of Tarski (see Axiom B7' in
footnote 1 of [KalishMontague] p.
81.) It is equivalent to axiom scheme
C10' in [Megill] p. 448 (p. 16 of the
preprint); the equivalence is
established by ax9o 1958 and ax9from9o 2232. A more convenient form of this
axiom is a9e 1956, which has additional remarks.
Raph Levien proved the independence of this axiom from the other logical
axioms on 12-Apr-2005. See item 16 at
http://us.metamath.org/award2003.html.
ax-9 1669 can be proved from the weaker version ax9v 1670
requiring that the
variables be distinct; see theorem ax9 1957.
ax-9 1669 can also be proved from the Axiom of
Separation (in the form that
we use that axiom, where free variables are not universally quantified).
See theorem ax9vsep 4365.
Except by ax9v 1670, this axiom should not be referenced
directly. Instead,
use theorem ax9 1957. (Contributed by NM, 5-Aug-1993.)
(New usage is discouraged.) |