Theorem ax6 2416

Description: Theorem showing that ax-6 1997 follows from the weaker version ax6v 1998. (Even though this theorem depends on ax-6 1997, all references of ax-6 1997 are made via ax6v 1998. An earlier version stated ax6v 1998 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 1997 so that all proofs can be traced back to ax6v 1998. When possible, use the weaker ax6v 1998 rather than ax6 2416 since the ax6v 1998 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 1998 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 1810	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
3	1, 2	mpbi 233	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1568  ∃wex 1809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-12 2213  ax-13 2404

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810

This theorem is referenced by:  axc10  2417