Theorem ax467to6 2171

Description: Re-derivation of ax-6o 2137 from ax467 2169. Note that ax-6o 2137 and ax-7 1734 are not used by the re-derivation. The use of alimi 1559 (which uses ax-4 2135) is allowed since we have already proved ax467to4 2170. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax467to6	⊢ (¬ ∀x ¬ ∀xφ → φ)

Proof of Theorem ax467to6

Step	Hyp	Ref	Expression
1		hba1-o 2149	. . . . . 6 ⊢ (∀xφ → ∀x∀xφ)
2	1	con3i 127	. . . . 5 ⊢ (¬ ∀x∀xφ → ¬ ∀xφ)
3	2	alimi 1559	. . . 4 ⊢ (∀x ¬ ∀x∀xφ → ∀x ¬ ∀xφ)
4	3	sps-o 2159	. . 3 ⊢ (∀x∀x ¬ ∀x∀xφ → ∀x ¬ ∀xφ)
5	4	con3i 127	. 2 ⊢ (¬ ∀x ¬ ∀xφ → ¬ ∀x∀x ¬ ∀x∀xφ)
6		pm2.21 100	. 2 ⊢ (¬ ∀x∀x ¬ ∀x∀xφ → (∀x∀x ¬ ∀x∀xφ → ∀xφ))
7		ax467 2169	. 2 ⊢ ((∀x∀x ¬ ∀x∀xφ → ∀xφ) → φ)
8	5, 6, 7	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:  ax467to7  2172