Theorem hbn 1776

Description: If x is not free in φ, it is not free in ¬ φ. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 17-Dec-2017.)

Hypothesis

Ref	Expression
hbn.1	⊢ (φ → ∀xφ)

Assertion

Ref	Expression
hbn	⊢ (¬ φ → ∀x ¬ φ)

Proof of Theorem hbn

Step	Hyp	Ref	Expression
1		hbnt 1775	. 2 ⊢ (∀x(φ → ∀xφ) → (¬ φ → ∀x ¬ φ))
2		hbn.1	. 2 ⊢ (φ → ∀xφ)
3	1, 2	mpg 1548	1 ⊢ (¬ 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:  hba1  1786  hbimOLD  1818  spimehOLD  1821  hbanOLD  1829  hbex  1841  cbv3hvOLD  1851  hbnae  1955  hbnae-o  2179