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Theorem ax6 2249
Description: Theorem showing that ax-6 1886 follows from the weaker version ax6v 1887. (Even though this theorem depends on ax-6 1886, all references of ax-6 1886 are made via ax6v 1887. An earlier version stated ax6v 1887 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 1886 so that all proofs can be traced back to ax6v 1887. When possible, use the weaker ax6v 1887 rather than ax6 2249 since the ax6v 1887 derivation is much shorter and requires fewer axioms. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 4-Feb-2018.)

Assertion
Ref Expression
ax6 ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Proof of Theorem ax6
StepHypRef Expression
1 ax6e 2248 . 2 𝑥 𝑥 = 𝑦
2 df-ex 1703 . 2 (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
31, 2mpbi 220 1 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1479  wex 1702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-12 2045  ax-13 2244
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703
This theorem is referenced by:  axc10  2250
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