Theorem ax6 2384

Description: Theorem showing that ax-6 1968 follows from the weaker version ax6v 1969. (Even though this theorem depends on ax-6 1968, all references of ax-6 1968 are made via ax6v 1969. An earlier version stated ax6v 1969 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 1968 so that all proofs can be traced back to ax6v 1969. When possible, use the weaker ax6v 1969 rather than ax6 2384 since the ax6v 1969 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 2372. Use ax6v 1969 instead. (New usage is discouraged.)

Assertion

Ref	Expression
ax6	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Proof of Theorem ax6

Step	Hyp	Ref	Expression
1		ax6e 2383	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		df-ex 1781	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
3	1, 2	mpbi 230	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1539  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-12 2180  ax-13 2372

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781

This theorem is referenced by:  axc10  2385