Theorem hbnt 710

Description: A closed form of hypothesis builder hbne 699.

Assertion

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

Proof of Theorem hbnt

Step	Hyp	Ref	Expression
1		con3 86	. . 3 ⊢ ((φ → ∀xφ) → (¬ ∀xφ → ¬ φ))
2	1	19.20ii 692	. 2 ⊢ (∀x(φ → ∀xφ) → (∀x ¬ ∀xφ → ∀x ¬ φ))
3		ax-6 675	. . 3 ⊢ (¬ ∀x ¬ ∀xφ → φ)
4	3	con1i 88	. 2 ⊢ (¬ φ → ∀x ¬ ∀xφ)
5	2, 4	syl5 22	1 ⊢ (∀x(φ → ∀xφ) → (¬ φ → ∀x ¬ φ))

Syntax hints:  ¬ wn 1   → wi 2  ∀wal 672

This theorem is referenced by:  19.9t 719  hbnd 786

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677

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