Theorem ax6 2418

Description: Theorem showing that ax-6 1990 follows from the weaker version ax6v 1991. (Even though this theorem depends on ax-6 1990, all references of ax-6 1990 are made via ax6v 1991. An earlier version stated ax6v 1991 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 1990 so that all proofs can be traced back to ax6v 1991. When possible, use the weaker ax6v 1991 rather than ax6 2418 since the ax6v 1991 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 2406. Use ax6v 1991 instead. (New usage is discouraged.)

Assertion

Ref	Expression
ax6	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Proof of Theorem ax6

Step	Hyp	Ref	Expression
1		ax6e 2417	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		df-ex 1803	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
3	1, 2	mpbi 233	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1561  ∃wex 1802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-12 2215  ax-13 2406

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803

This theorem is referenced by:  axc10  2419