Theorem hbnt 1775

Description: Closed theorem version of bound-variable hypothesis builder hbn 1776. (Contributed by NM, 5-Aug-1993.)

Assertion

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

Proof of Theorem hbnt

Step	Hyp	Ref	Expression
1		ax6o 1750	. . 3 ⊢ (¬ ∀x ¬ ∀xφ → φ)
2	1	con1i 121	. 2 ⊢ (¬ φ → ∀x ¬ ∀xφ)
3		con3 126	. . 3 ⊢ ((φ → ∀xφ) → (¬ ∀xφ → ¬ φ))
4	3	al2imi 1561	. 2 ⊢ (∀x(φ → ∀xφ) → (∀x ¬ ∀xφ → ∀x ¬ φ))
5	2, 4	syl5 28	1 ⊢ (∀x(φ → ∀xφ) → (¬ φ → ∀x ¬ φ))

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-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by:  hbn  1776  19.9ht  1778  nfnd  1791  nfimdOLD  1809  hbnd  1883