Theorem ax6 2392

Description: Theorem showing that ax-6 1974 follows from the weaker version ax6v 1975. (Even though this theorem depends on ax-6 1974, all references of ax-6 1974 are made via ax6v 1975. An earlier version stated ax6v 1975 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 1974 so that all proofs can be traced back to ax6v 1975. When possible, use the weaker ax6v 1975 rather than ax6 2392 since the ax6v 1975 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 2380. Use ax6v 1975 instead. (New usage is discouraged.)

Assertion

Ref	Expression
ax6	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Proof of Theorem ax6

Step	Hyp	Ref	Expression
1		ax6e 2391	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		df-ex 1787	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
3	1, 2	mpbi 231	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1545  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-12 2189  ax-13 2380

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787

This theorem is referenced by:  axc10  2393