Theorem ax6 2386

Description: Theorem showing that ax-6 1964 follows from the weaker version ax6v 1965. (Even though this theorem depends on ax-6 1964, all references of ax-6 1964 are made via ax6v 1965. An earlier version stated ax6v 1965 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 1964 so that all proofs can be traced back to ax6v 1965. When possible, use the weaker ax6v 1965 rather than ax6 2386 since the ax6v 1965 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 2374. Use ax6v 1965 instead. (New usage is discouraged.)

Assertion

Ref	Expression
ax6	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Proof of Theorem ax6

Step	Hyp	Ref	Expression
1		ax6e 2385	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		df-ex 1776	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
3	1, 2	mpbi 230	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1534  ∃wex 1775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-12 2174  ax-13 2374

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776

This theorem is referenced by:  axc10  2387