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Theorem ax6v 2058
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 2057 by adding a distinct variable restriction ($d). From here on, ax-6 2057 should not be referenced directly by any other proof, so that theorem ax6 2413 will show that we can recover ax-6 2057 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 2058 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 2057.

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
StepHypRef Expression
1 ax-6 2057 1 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1629
This theorem was proved from axioms:  ax-6 2057
This theorem is referenced by:  ax6ev  2059  spimw  2084  bj-denot  32998  bj-axc10v  33053  axc5c4c711toc5  39129
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