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Theorem ax6v 1839
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 1838 by adding a distinct variable restriction ($d). From here on, ax-6 1838 should not be referenced directly by any other proof, so that theorem ax6 2142 will show that we can recover ax-6 1838 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 1839 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 1838.

When possible, use of this theorem rather than ax6 2142 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
StepHypRef Expression
1 ax-6 1838 1 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1472
This theorem was proved from axioms:  ax-6 1838
This theorem is referenced by:  ax6ev  1840  spimw  1876  bj-denot  31684  bj-axc10v  31739  axc5c4c711toc5  37507
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