**Description: **Axiom B7 of [Tarski] p. 75, which requires that 𝑥 and
𝑦
be
distinct. This trivial proof is intended merely to weaken axiom ax-6 2021
by adding a distinct variable restriction ($d). From here on, ax-6 2021
should not be referenced directly by any other proof, so that theorem
ax6 2348 will show that we can recover ax-6 2021
from this weaker version if
it were an axiom (as it is in the case of Tarski).
Note: Introducing 𝑥, 𝑦 as a distinct variable group
"out of the
blue" with no apparent justification has puzzled some people, but
it is
perfectly sound. All we are doing is adding an additional prerequisite,
similar to adding an unnecessary logical hypothesis, that results in a
weakening of the theorem. This means that any *future* theorem
that
references ax6v 2022 must have a $d specified for the two
variables that
get substituted for 𝑥 and 𝑦. The $d does not
propagate
"backwards", i.e., it does not impose a requirement on ax-6 2021.
When possible, use of this theorem rather than ax6 2348 is
preferred since
its derivation is much shorter and requires fewer axioms. (Contributed
by NM, 7-Aug-2015.) |