Axiom ax-6 675

Description: Axiom of Quantified Negation. This axiom is used to manipulate negated quantifiers. One of the 4 axioms of pure predicate calculus. Equivalent to axiom scheme C7' in [Megill] p. 448 (p. 16 of the preprint). Another equivalent variant ax6 711 appears as Axiom C5-2 of [Monk2] p. 113.

Assertion

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

Detailed syntax breakdown of Axiom ax-6

Step	Hyp	Ref	Expression
1		wph	. . . . . 6 wff φ
2		vx	. . . . . 6 set x
3	1, 2	wal 672	. . . . 5 wff ∀xφ
4	3	wn 1	. . . 4 wff ¬ ∀xφ
5	4, 2	wal 672	. . 3 wff ∀x ¬ ∀xφ
6	5	wn 1	. 2 wff ¬ ∀x ¬ ∀xφ
7	6, 1	wi 2	1 wff (¬ ∀x ¬ ∀xφ → φ)

This axiom is referenced by:  a6e 688  hbne 699  hbnt 710  19.9r 718  ax9a 808  eqid 810

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