Theorem hbn 1678

Description: If 𝑥 is not free in 𝜑, it is not free in ¬ 𝜑. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
hbn.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
hbn	⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)

Proof of Theorem hbn

Step	Hyp	Ref	Expression
1		hbnt 1677	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
2		hbn.1	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3	1, 2	mpg 1475	1 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1471  ax-gen 1473  ax-ie2 1518  ax-4 1534  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379

This theorem is referenced by:  hbnae  1745  sbn  1981  euor  2081  euor2  2113