Theorem ax6 2142

Description: Theorem showing that ax-6 1836 follows from the weaker version ax6v 1837. (Even though this theorem depends on ax-6 1836, all references of ax-6 1836 are made via ax6v 1837. An earlier version stated ax6v 1837 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 1836 so that all proofs can be traced back to ax6v 1837. (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 2141	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		df-ex 1693	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
3	1, 2	mpbi 215	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1466  ∃wex 1692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-12 1983  ax-13 2137

This theorem depends on definitions:  df-bi 192  df-an 380  df-ex 1693

This theorem is referenced by:  axc10  2143