Theorem ax6 983

Description: Rederivation of axiom ax-6 965 from the orginal version, ax-6o 982. See ax6o 981 for the derivation of ax-6o 982 from ax-6 965.

This theorem should not be referenced in any proof. Instead, use ax-6 965 above so that uses of ax-6 965 can be more easily identified.

Assertion

Ref	Expression
ax6	⊢ (¬ ∀xφ → ∀x ¬ ∀xφ)

Proof of Theorem ax6

Step	Hyp	Ref	Expression
1		ax-4 977	. . . . 5 ⊢ (∀x ¬ ∀x∀xφ → ¬ ∀x∀xφ)
2		id 59	. . . . . . 7 ⊢ (∀xφ → ∀xφ)
3	2	ax-gen 967	. . . . . 6 ⊢ ∀x(∀xφ → ∀xφ)
4		ax-5o 979	. . . . . 6 ⊢ (∀x(∀xφ → ∀xφ) → (∀xφ → ∀x∀xφ))
5	3, 4	ax-mp 7	. . . . 5 ⊢ (∀xφ → ∀x∀xφ)
6	1, 5	nsyl 116	. . . 4 ⊢ (∀x ¬ ∀x∀xφ → ¬ ∀xφ)
7	6	ax-gen 967	. . 3 ⊢ ∀x(∀x ¬ ∀x∀xφ → ¬ ∀xφ)
8		ax-5o 979	. . 3 ⊢ (∀x(∀x ¬ ∀x∀xφ → ¬ ∀xφ) → (∀x ¬ ∀x∀xφ → ∀x ¬ ∀xφ))
9	7, 8	ax-mp 7	. 2 ⊢ (∀x ¬ ∀x∀xφ → ∀x ¬ ∀xφ)
10		ax-6o 982	. 2 ⊢ (¬ ∀x ¬ ∀x∀xφ → ∀xφ)
11	9, 10	nsyl4 120	1 ⊢ (¬ ∀xφ → ∀x ¬ ∀xφ)

Syntax hints:  ¬ wn 2   → wi 3  ∀wal 958

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 967  ax-4 977  ax-5o 979  ax-6o 982

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