Theorem ax6 2382

Description: Theorem showing that ax-6 1970 follows from the weaker version ax6v 1971. (Even though this theorem depends on ax-6 1970, all references of ax-6 1970 are made via ax6v 1971. An earlier version stated ax6v 1971 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 1970 so that all proofs can be traced back to ax6v 1971. When possible, use the weaker ax6v 1971 rather than ax6 2382 since the ax6v 1971 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 2370. Use ax6v 1971 instead. (New usage is discouraged.)

Assertion

Ref	Expression
ax6	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Proof of Theorem ax6

Step	Hyp	Ref	Expression
1		ax6e 2381	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		df-ex 1781	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
3	1, 2	mpbi 229	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1538  ∃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 1912  ax-6 1970  ax-7 2010  ax-12 2170  ax-13 2370

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1781

This theorem is referenced by:  axc10  2383