Theorem ax6 2413

Description: Theorem showing that ax-6 1999 follows from the weaker version ax6v 2000. (Even though this theorem depends on ax-6 1999, all references of ax-6 1999 are made via ax6v 2000. An earlier version stated ax6v 2000 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 1999 so that all proofs can be traced back to ax6v 2000. When possible, use the weaker ax6v 2000 rather than ax6 2413 since the ax6v 2000 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.) Usage of this theorem is discouraged because it depends on ax-13 2401. Use ax6v 2000 instead. (New usage is discouraged.)

Assertion

Ref	Expression
ax6	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Proof of Theorem ax6

Step	Hyp	Ref	Expression
1		ax6e 2412	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		df-ex 1812	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
3	1, 2	mpbi 233	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

This proof depends on syntax axioms:  ¬ wn 3  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-12 2212  ax-13 2401

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812

This theorem is used by:  axc10  2414