This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Description: Closed form of nexdh (actually, its general instance). (Contributed by BJ, 6-May-2019)
|
|
Ref |
Expression |
|
Assertion |
bj-nexdh |
⊢ ( ∀ 𝑥 ( 𝜑 → ¬ 𝜓 ) → ( ( 𝜒 → ∀ 𝑥 𝜑 ) → ( 𝜒 → ¬ ∃ 𝑥 𝜓 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
sylgt |
⊢ ( ∀ 𝑥 ( 𝜑 → ¬ 𝜓 ) → ( ( 𝜒 → ∀ 𝑥 𝜑 ) → ( 𝜒 → ∀ 𝑥 ¬ 𝜓 ) ) ) |
| 2 |
|
alnex |
⊢ ( ∀ 𝑥 ¬ 𝜓 ↔ ¬ ∃ 𝑥 𝜓 ) |
| 3 |
1 2
|
syl8ib |
⊢ ( ∀ 𝑥 ( 𝜑 → ¬ 𝜓 ) → ( ( 𝜒 → ∀ 𝑥 𝜑 ) → ( 𝜒 → ¬ ∃ 𝑥 𝜓 ) ) ) |