**Description: **Axiom of Existence. One
of the equality and substitution axioms of
predicate calculus with equality. This axiom tells us 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 axc10 2396 and ax6fromc10 36014. A more convenient form of this
axiom is ax6e 2394, 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
https://us.metamath.org/award2003.html 2394.
ax-6 1963 can be proved from the weaker version ax6v 1964
requiring that the
variables be distinct; see theorem ax6 2395.
ax-6 1963 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 ax6vsep 5198.
Except by ax6v 1964, this axiom should not be referenced
directly. Instead,
use theorem ax6 2395. (Contributed by NM, 10-Jan-1993.)
(New usage is discouraged.) |