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Theorem ax6 2404
Description: Theorem showing that ax-6 1971 follows from the weaker version ax6v 1972. (Even though this theorem depends on ax-6 1971, all references of ax-6 1971 are made via ax6v 1972. An earlier version stated ax6v 1972 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 1971 so that all proofs can be traced back to ax6v 1972. Usage of this theorem is discouraged because it depends on ax-13 2392. When possible, use the weaker ax6v 1972 rather than ax6 2404 since the ax6v 1972 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.) (New usage is discouraged.)

Assertion
Ref Expression
ax6 ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Proof of Theorem ax6
StepHypRef Expression
1 ax6e 2403 . 2 𝑥 𝑥 = 𝑦
2 df-ex 1782 . 2 (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
31, 2mpbi 233 1 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1536  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-12 2179  ax-13 2392
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782
This theorem is referenced by:  axc10  2405
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