Axiom ax-6 1729

Description: Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax6w 1717) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
ax-6	⊢ (¬ ∀xφ → ∀x ¬ ∀xφ)

Detailed syntax breakdown of Axiom ax-6

Step	Hyp	Ref	Expression
1		wph	. . . 4 wff φ
2		vx	. . . 4 setvar x
3	1, 2	wal 1540	. . 3 wff ∀xφ
4	3	wn 3	. 2 wff ¬ ∀xφ
5	4, 2	wal 1540	. 2 wff ∀x ¬ ∀xφ
6	4, 5	wi 4	1 wff (¬ ∀xφ → ∀x ¬ ∀xφ)

This axiom is referenced by:  hbn1  1730  ax6o  1750