Theorem ax67to6 1019

Description: Re-derivation of ax-6o 976 from ax67 1018. Note that ax-6o 976 and ax-7 960 are not used by the re-derivation.

Assertion

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

Proof of Theorem ax67to6

Step	Hyp	Ref	Expression
1		hba1 1001	. . . . 5 ⊢ (∀xφ → ∀x∀xφ)
2	1	con3i 98	. . . 4 ⊢ (¬ ∀x∀xφ → ¬ ∀xφ)
3	2	19.20i 990	. . 3 ⊢ (∀x ¬ ∀x∀xφ → ∀x ¬ ∀xφ)
4	3	con3i 98	. 2 ⊢ (¬ ∀x ¬ ∀xφ → ¬ ∀x ¬ ∀x∀xφ)
5		ax67 1018	. 2 ⊢ (¬ ∀x ¬ ∀x∀xφ → ∀xφ)
6		ax-4 971	. 2 ⊢ (∀xφ → φ)
7	4, 5, 6	3syl 20	1 ⊢ (¬ ∀x ¬ ∀xφ → φ)

Syntax hints:  ¬ wn 2   → wi 3  ∀wal 952

This theorem is referenced by:  ax67to7 1020

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-4 971  ax-5o 973  ax-6o 976

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