![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nfntht2 | Structured version Visualization version GIF version |
Description: Closed form of nfnth 1799. (Contributed by BJ, 16-Sep-2021.) (Proof shortened by Wolf Lammen, 4-Sep-2022.) |
Ref | Expression |
---|---|
nfntht2 | ⊢ (∀𝑥 ¬ 𝜑 → Ⅎ𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alnex 1778 | . 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
2 | nfntht 1790 | . 2 ⊢ (¬ ∃𝑥𝜑 → Ⅎ𝑥𝜑) | |
3 | 1, 2 | sylbi 217 | 1 ⊢ (∀𝑥 ¬ 𝜑 → Ⅎ𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1535 ∃wex 1776 Ⅎwnf 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 207 df-ex 1777 df-nf 1781 |
This theorem is referenced by: nfnth 1799 |
Copyright terms: Public domain | W3C validator |