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Metamath Proof Explorer
Theorem hbn
Description: If x is not free in ph , it is not free in -. ph .
(Contributed by NM, 10-Jan-1993) (Proof shortened by Wolf Lammen, 17-Dec-2017)
|
|
Ref |
Expression |
|
Hypothesis |
hbn.1 |
⊢ ( 𝜑 → ∀ 𝑥 𝜑 ) |
|
Assertion |
hbn |
⊢ ( ¬ 𝜑 → ∀ 𝑥 ¬ 𝜑 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
hbn.1 |
⊢ ( 𝜑 → ∀ 𝑥 𝜑 ) |
| 2 |
|
hbnt |
⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) → ( ¬ 𝜑 → ∀ 𝑥 ¬ 𝜑 ) ) |
| 3 |
2 1
|
mpg |
⊢ ( ¬ 𝜑 → ∀ 𝑥 ¬ 𝜑 ) |