**Description: **Axiom of Equality. One
of the equality and substitution axioms of
predicate calculus with equality. This is similar to, but not quite, a
transitive law for equality (proved later as equtr 1685). Axiom scheme C8'
in [Megill] p. 448 (p. 16 of the preprint).
Also appears as Axiom C7 of
[Monk2] p. 105.
Axioms ax-8 1482 through ax-16 1786 are the axioms having to do with equality,
substitution, and logical properties of our binary predicate ∈ (which
later in set theory will mean "is a member of"). Note that all
axioms
except ax-16 1786 and ax-17 1506 are still valid even when 𝑥, 𝑦, and
𝑧 are replaced with the same variable
because they do not have any
distinct variable (Metamath's $d) restrictions. Distinct variable
restrictions are required for ax-16 1786 and ax-17 1506 only. (Contributed by
NM, 5-Aug-1993.) |