Axiom ax-10 2156

Description: Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax10w 2143) 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 2157 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 2156 through ax-13 2379, by invoking ax10w 2143 through ax13w 2150. We encourage proving theorems *without* ax-10 2156 through ax-13 2379 and moving them up to the ax-4 1874 through ax-9 2136 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 1618	. . 3 wff ∀𝑥𝜑
4	3	wn 3	. 2 wff ¬ ∀𝑥𝜑
5	4, 2	wal 1618	. 2 wff ∀𝑥 ¬ ∀𝑥𝜑
6	4, 5	wi 4	1 wff (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

This axiom is referenced by:  hbn1  2157