Theorem ax6 2147

Description: Rederivation of Axiom ax-6 1729 from ax-6o 2137 and other older axioms. See ax6o 1750 for the derivation of ax-6o 2137 from ax-6 1729. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem ax6

Step	Hyp	Ref	Expression
1		ax-5o 2136	. . 3 ⊢ (∀x(∀x ¬ ∀x∀xφ → ¬ ∀xφ) → (∀x ¬ ∀x∀xφ → ∀x ¬ ∀xφ))
2		ax-4 2135	. . . 4 ⊢ (∀x ¬ ∀x∀xφ → ¬ ∀x∀xφ)
3		ax-5o 2136	. . . . 5 ⊢ (∀x(∀xφ → ∀xφ) → (∀xφ → ∀x∀xφ))
4		id 19	. . . . 5 ⊢ (∀xφ → ∀xφ)
5	3, 4	mpg 1548	. . . 4 ⊢ (∀xφ → ∀x∀xφ)
6	2, 5	nsyl 113	. . 3 ⊢ (∀x ¬ ∀x∀xφ → ¬ ∀xφ)
7	1, 6	mpg 1548	. 2 ⊢ (∀x ¬ ∀x∀xφ → ∀x ¬ ∀xφ)
8		ax-6o 2137	. 2 ⊢ (¬ ∀x ¬ ∀x∀xφ → ∀xφ)
9	7, 8	nsyl4 134	1 ⊢ (¬ ∀xφ → ∀x ¬ ∀xφ)

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-4 2135  ax-5o 2136  ax-6o 2137

This theorem is referenced by:  hba1-o  2149  ax467  2169  equidq  2175