Axiom ax-10 2006

Description: Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax10w 1993) but is used as an auxiliary axiom scheme to achieve scheme completeness. It means that 𝑥 is not free in ¬ ∀𝑥𝜑. (Contributed by NM, 21-May-2008.) Use its alias hbn1 2007 instead if you must use it. Any theorem in first order logic (FOL) that contains only set variables that are all mutually distinct, and has no wff variables, can be proved *without* using ax-10 2006 through ax-13 2234, by invoking ax10w 1993 through ax13w 2000. We encourage proving theorems *without* ax-10 2006 through ax-13 2234 and moving them up to the ax-4 1728 through ax-9 1986 section. (New usage is discouraged.)

Assertion

Ref	Expression
ax-10	⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Detailed syntax breakdown of Axiom ax-10

Step	Hyp	Ref	Expression
1		wph	. . . 4 wff 𝜑
2		vx	. . . 4 setvar 𝑥
3	1, 2	wal 1473	. . 3 wff ∀𝑥𝜑
4	3	wn 3	. 2 wff ¬ ∀𝑥𝜑
5	4, 2	wal 1473	. 2 wff ∀𝑥 ¬ ∀𝑥𝜑
6	4, 5	wi 4	1 wff (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

This axiom is referenced by:  hbn1  2007