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Theorem ax6v 1837
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 1836 by adding a distinct variable restriction. From here on, ax-6 1836 should not be referenced directly by any other proof, so that theorem ax6 2142 will show that we can recover ax-6 1836 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 redundant requirement, no different from adding a redundant logical hypothesis, that results in a weakening of the theorem. This means that any future theorem that references ax6v 1837 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 1836.

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 1836 1 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1466
This theorem was proved from axioms:  ax-6 1836
This theorem is referenced by:  ax6ev  1838  spimw  1874  bj-denot  31455  bj-axc10v  31505  axc5c4c711toc5  37110
  Copyright terms: Public domain W3C validator