Theorem ax6v 2000

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 1999 by adding a distinct variable restriction ($d). From here on, ax-6 1999 should not be referenced directly by any other proof, so that Theorem ax6 2413 will show that we can recover ax-6 1999 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 2000 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 1999.

When possible, use of this theorem rather than ax6 2413 is preferred since its derivation is much shorter and requires fewer axioms. (Contributed by NM, 7-Aug-2015.)

Assertion

Ref	Expression
ax6v	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem ax6v

Step	Hyp	Ref	Expression
1		ax-6 1999	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

This proof depends on syntax axioms:  ¬ wn 3  ∀wal 1567

This proof depends on axioms:  ax-6 1999

This theorem is used by:  ax6ev  2001  spimw  2002  spimew  2003  iresn0n0  6045  bj-denot  37667  bj-axc10v  37798  axc5c4c711toc5  45480