Theorem ax6 2414

Description: Theorem showing that ax-6 1986 follows from the weaker version ax6v 1987. (Even though this theorem depends on ax-6 1986, all references of ax-6 1986 are made via ax6v 1987. An earlier version stated ax6v 1987 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 1986 so that all proofs can be traced back to ax6v 1987. When possible, use the weaker ax6v 1987 rather than ax6 2414 since the ax6v 1987 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 2402. Use ax6v 1987 instead. (New usage is discouraged.)

Assertion

Ref	Expression
ax6	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Proof of Theorem ax6

Step	Hyp	Ref	Expression
1		ax6e 2413	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		df-ex 1799	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
3	1, 2	mpbi 232	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1557  ∃wex 1798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-12 2211  ax-13 2402

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799

This theorem is referenced by:  axc10  2415