Theorem ax467to6 1021

Description: Re-derivation of ax-6o 975 from ax467 1019. Note that ax-6o 975 and ax-7 959 are not used by the re-derivation. The use of 19.20i 989 (which uses ax-4 970) is allowed since we have already proved ax467to4 1020.

Assertion

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

Proof of Theorem ax467to6

Step	Hyp	Ref	Expression
1		ax467to4 1020	. . . 4 ⊢ (∀x∀x ¬ ∀x∀xφ → ∀x ¬ ∀x∀xφ)
2		hba1 1000	. . . . . 6 ⊢ (∀xφ → ∀x∀xφ)
3	2	con3i 98	. . . . 5 ⊢ (¬ ∀x∀xφ → ¬ ∀xφ)
4	3	19.20i 989	. . . 4 ⊢ (∀x ¬ ∀x∀xφ → ∀x ¬ ∀xφ)
5	1, 4	syl 10	. . 3 ⊢ (∀x∀x ¬ ∀x∀xφ → ∀x ¬ ∀xφ)
6	5	con3i 98	. 2 ⊢ (¬ ∀x ¬ ∀xφ → ¬ ∀x∀x ¬ ∀x∀xφ)
7		pm2.21 76	. 2 ⊢ (¬ ∀x∀x ¬ ∀x∀xφ → (∀x∀x ¬ ∀x∀xφ → ∀xφ))
8		ax467 1019	. 2 ⊢ ((∀x∀x ¬ ∀x∀xφ → ∀xφ) → φ)
9	6, 7, 8	3syl 20	1 ⊢ (¬ ∀x ¬ ∀xφ → φ)

Syntax hints:  ¬ wn 2   → wi 3  ∀wal 951

This theorem is referenced by:  ax467to7 1022

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-4 970  ax-5o 972  ax-6o 975

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