Theorem ax6o 1750

Description: Show that the original axiom ax-6o 2137 can be derived from ax-6 1729 and others. See ax6 2147 for the rederivation of ax-6 1729 from ax-6o 2137.

Normally, ax6o 1750 should be used rather than ax-6o 2137, except by theorems specifically studying the latter's properties. (Contributed by NM, 21-May-2008.)

Assertion

Ref	Expression
ax6o	⊢ (¬ ∀x ¬ ∀xφ → φ)

Proof of Theorem ax6o

Step	Hyp	Ref	Expression
1		sp 1747	. 2 ⊢ (∀xφ → φ)
2		ax-6 1729	. 2 ⊢ (¬ ∀xφ → ∀x ¬ ∀xφ)
3	1, 2	nsyl4 134	1 ⊢ (¬ ∀x ¬ ∀xφ → φ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by:  a6e  1751  modal-b  1752  hbnt  1775  nfndOLD  1792  equsalhwOLD  1839  ax9o  1950