Theorem ax6 2404

Description: Theorem showing that ax-6 2077 follows from the weaker version ax6v 2078. (Even though this theorem depends on ax-6 2077, all references of ax-6 2077 are made via ax6v 2078. An earlier version stated ax6v 2078 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 2077 so that all proofs can be traced back to ax6v 2078. When possible, use the weaker ax6v 2078 rather than ax6 2404 since the ax6v 2078 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 2403	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		df-ex 1881	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
3	1, 2	mpbi 222	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1656  ∃wex 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-12 2222  ax-13 2390

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1881

This theorem is referenced by:  axc10  2405