Theorem ax6 2251

Description: Theorem showing that ax-6 1888 follows from the weaker version ax6v 1889. (Even though this theorem depends on ax-6 1888, all references of ax-6 1888 are made via ax6v 1889. An earlier version stated ax6v 1889 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 1888 so that all proofs can be traced back to ax6v 1889. When possible, use the weaker ax6v 1889 rather than ax6 2251 since the ax6v 1889 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 2250	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		df-ex 1705	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
3	1, 2	mpbi 220	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  axc10  2252