Theorem ax67to6 2167

Description: Re-derivation of ax-6o 2137 from ax67 2165. Note that ax-6o 2137 and ax-7 1734 are not used by the re-derivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax67to6	⊢ (¬ ∀x ¬ ∀xφ → φ)

Proof of Theorem ax67to6

Step	Hyp	Ref	Expression
1		hba1-o 2149	. . . . 5 ⊢ (∀xφ → ∀x∀xφ)
2	1	con3i 127	. . . 4 ⊢ (¬ ∀x∀xφ → ¬ ∀xφ)
3	2	alimi 1559	. . 3 ⊢ (∀x ¬ ∀x∀xφ → ∀x ¬ ∀xφ)
4	3	con3i 127	. 2 ⊢ (¬ ∀x ¬ ∀xφ → ¬ ∀x ¬ ∀x∀xφ)
5		ax67 2165	. 2 ⊢ (¬ ∀x ¬ ∀x∀xφ → ∀xφ)
6		ax-4 2135	. 2 ⊢ (∀xφ → φ)
7	4, 5, 6	3syl 18	1 ⊢ (¬ ∀x ¬ ∀xφ → φ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-7 1734  ax-4 2135  ax-5o 2136  ax-6o 2137

This theorem is referenced by:  ax67to7  2168