MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax6v Structured version   Visualization version   GIF version

Theorem ax6v 1886
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 1885 by adding a distinct variable restriction ($d). From here on, ax-6 1885 should not be referenced directly by any other proof, so that theorem ax6 2250 will show that we can recover ax-6 1885 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 1886 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 1885.

When possible, use of this theorem rather than ax6 2250 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 1885 1 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1478
This theorem was proved from axioms:  ax-6 1885
This theorem is referenced by:  ax6ev  1887  spimw  1923  bj-denot  32301  bj-axc10v  32356  axc5c4c711toc5  38082
  Copyright terms: Public domain W3C validator