Theorem ax6 2383

Description: Theorem showing that ax-6 1971 follows from the weaker version ax6v 1972. (Even though this theorem depends on ax-6 1971, all references of ax-6 1971 are made via ax6v 1972. An earlier version stated ax6v 1972 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 1971 so that all proofs can be traced back to ax6v 1972. When possible, use the weaker ax6v 1972 rather than ax6 2383 since the ax6v 1972 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 2371. Use ax6v 1972 instead. (New usage is discouraged.)

Assertion

Ref	Expression
ax6	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Proof of Theorem ax6

Step	Hyp	Ref	Expression
1		ax6e 2382	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		df-ex 1782	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
3	1, 2	mpbi 229	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1539  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171  ax-13 2371

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782

This theorem is referenced by:  axc10  2384