Theorem ax6 2419

Description: Theorem showing that ax-6 2000 follows from the weaker version ax6v 2001. (Even though this theorem depends on ax-6 2000, all references of ax-6 2000 are made via ax6v 2001. An earlier version stated ax6v 2001 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 2000 so that all proofs can be traced back to ax6v 2001. When possible, use the weaker ax6v 2001 rather than ax6 2419 since the ax6v 2001 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 2407. Use ax6v 2001 instead. (New usage is discouraged.)

Assertion

Ref	Expression
ax6	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Proof of Theorem ax6

Step	Hyp	Ref	Expression
1		ax6e 2418	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		df-ex 1813	. 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 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2216  ax-13 2407

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by:  axc10  2420