Theorem ax6 2255

Description: Theorem showing that ax-6 1890 follows from the weaker version ax6v 1891. (Even though this theorem depends on ax-6 1890, all references of ax-6 1890 are made via ax6v 1891. An earlier version stated ax6v 1891 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 1890 so that all proofs can be traced back to ax6v 1891. When possible, use the weaker ax6v 1891 rather than ax6 2255 since the ax6v 1891 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 2254	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		df-ex 1702	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
3	1, 2	mpbi 220	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1478  ∃wex 1701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-12 2049  ax-13 2250

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702

This theorem is referenced by:  axc10  2256