Theorem ax6 2389

Description: Theorem showing that ax-6 1969 follows from the weaker version ax6v 1970. (Even though this theorem depends on ax-6 1969, all references of ax-6 1969 are made via ax6v 1970. An earlier version stated ax6v 1970 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 1969 so that all proofs can be traced back to ax6v 1970. When possible, use the weaker ax6v 1970 rather than ax6 2389 since the ax6v 1970 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 2377. Use ax6v 1970 instead. (New usage is discouraged.)

Assertion

Ref	Expression
ax6	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Proof of Theorem ax6

Step	Hyp	Ref	Expression
1		ax6e 2388	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		df-ex 1782	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
3	1, 2	mpbi 230	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1540  ∃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 1912  ax-6 1969  ax-7 2010  ax-12 2185  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782

This theorem is referenced by:  axc10  2390