Theorem ax6 2402

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. Usage of this theorem is discouraged because it depends on ax-13 2390. When possible, use the weaker ax6v 1971 rather than ax6 2402 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.) (New usage is discouraged.)

Assertion

Ref	Expression
ax6	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Proof of Theorem ax6

Step	Hyp	Ref	Expression
1		ax6e 2401	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		df-ex 1781	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
3	1, 2	mpbi 232	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1535  ∃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 1970  ax-7 2015  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781

This theorem is referenced by:  axc10  2403