Theorem ax6 2287

Description: Theorem showing that ax-6 1945 follows from the weaker version ax6v 1946. (Even though this theorem depends on ax-6 1945, all references of ax-6 1945 are made via ax6v 1946. An earlier version stated ax6v 1946 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 1945 so that all proofs can be traced back to ax6v 1946. When possible, use the weaker ax6v 1946 rather than ax6 2287 since the ax6v 1946 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.)

Assertion

Ref	Expression
ax6	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Proof of Theorem ax6

Step	Hyp	Ref	Expression
1		ax6e 2286	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		df-ex 1745	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
3	1, 2	mpbi 220	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1521  ∃wex 1744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087  ax-13 2282

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745

This theorem is referenced by:  axc10  2288