Theorem hbn 2283

Description: If 𝑥 is not free in 𝜑, it is not free in ¬ 𝜑. (Contributed by NM, 10-Jan-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 2282	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
2		hbn.1	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3	1, 2	mpg 1791	1 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2163

This theorem depends on definitions:  df-bi 206  df-or 845  df-ex 1774  df-nf 1778

This theorem is referenced by:  hbnae  2423  ac6s6  37530  hbnae-o  38288  vk15.4j  43778  vk15.4jVD  44164