Theorem ax6 2416

Description: Theorem showing that ax-6 2002 follows from the weaker version ax6v 2003. (Even though this theorem depends on ax-6 2002, all references of ax-6 2002 are made via ax6v 2003. An earlier version stated ax6v 2003 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 2002 so that all proofs can be traced back to ax6v 2003. When possible, use the weaker ax6v 2003 rather than ax6 2416 since the ax6v 2003 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 2404. Use ax6v 2003 instead. (New usage is discouraged.)

Assertion

Ref	Expression
ax6	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Proof of Theorem ax6

Step	Hyp	Ref	Expression
1		ax6e 2415	. 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 1945  ax-6 2002  ax-7 2043  ax-12 2216  ax-13 2404

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

This theorem is used by:  axc10  2417