Theorem ax6 2239

Description: Theorem showing that ax-6 1875 follows from the weaker version ax6v 1876. (Even though this theorem depends on ax-6 1875, all references of ax-6 1875 are made via ax6v 1876. An earlier version stated ax6v 1876 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 1875 so that all proofs can be traced back to ax6v 1876. When possible, use the weaker ax6v 1876 rather than ax6 2239 since the ax6v 1876 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 2238	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		df-ex 1696	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
3	1, 2	mpbi 219	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by:  axc10  2240