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Theorem ax6 2237
Description: Theorem showing that ax-6 1874 follows from the weaker version ax6v 1875. (Even though this theorem depends on ax-6 1874, all references of ax-6 1874 are made via ax6v 1875. An earlier version stated ax6v 1875 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 1874 so that all proofs can be traced back to ax6v 1875. When possible, use the weaker ax6v 1875 rather than ax6 2237 since the ax6v 1875 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 2236 . 2 𝑥 𝑥 = 𝑦
2 df-ex 1695 . 2 (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
31, 2mpbi 218 1 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1472  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2032  ax-13 2232
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695
This theorem is referenced by:  axc10  2238
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