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Theorem ax6 2413
Description: Theorem showing that ax-6 2057 follows from the weaker version ax6v 2058. (Even though this theorem depends on ax-6 2057, all references of ax-6 2057 are made via ax6v 2058. An earlier version stated ax6v 2058 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 2057 so that all proofs can be traced back to ax6v 2058. When possible, use the weaker ax6v 2058 rather than ax6 2413 since the ax6v 2058 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 2412 . 2 𝑥 𝑥 = 𝑦
2 df-ex 1853 . 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 1629  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-12 2203  ax-13 2408
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853
This theorem is referenced by:  axc10  2414
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