Axiom ax-c7 38357

Description: Axiom of Quantified Negation. This axiom is used to manipulate negated quantifiers. Equivalent to axiom scheme C7' in [Megill] p. 448 (p. 16 of the preprint). An alternate axiomatization could use axc5c711 38390 in place of ax-c5 38355, ax-c7 38357, and ax-11 2147.

This axiom is obsolete and should no longer be used. It is proved above as Theorem axc7 2306. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-c7	⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

Detailed syntax breakdown of Axiom ax-c7

Step	Hyp	Ref	Expression
1		wph	. . . . . 6 wff 𝜑
2		vx	. . . . . 6 setvar 𝑥
3	1, 2	wal 1532	. . . . 5 wff ∀𝑥𝜑
4	3	wn 3	. . . 4 wff ¬ ∀𝑥𝜑
5	4, 2	wal 1532	. . 3 wff ∀𝑥 ¬ ∀𝑥𝜑
6	5	wn 3	. 2 wff ¬ ∀𝑥 ¬ ∀𝑥𝜑
7	6, 1	wi 4	1 wff (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)

This axiom is referenced by:  ax10fromc7  38367  ax6fromc10  38368  equid1  38371  axc5c7  38383  axc711  38386  axc5c711  38390  equid1ALT  38397