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| Mirrors > Home > MPE Home > Th. List > nfntht2 | Structured version Visualization version GIF version | ||
| Description: Closed form of nfnth 1801. (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 1780 | . 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
| 2 | nfntht 1792 | . 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 1537 ∃wex 1778 Ⅎwnf 1782 | 
| 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 1779 df-nf 1783 | 
| This theorem is referenced by: nfnth 1801 | 
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