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 by adding a distinct variable restriction ($d). From here on, ax-6 should not be referenced directly by any other proof, so that theorem
ax6 will show that we can recover ax-6 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 anyfuture theorem that
references ax6v 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 .

When possible, use of this theorem rather than ax6 is preferred since
its derivation is much shorter and requires fewer axioms. (Contributed by NM, 7-Aug-2015)