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| Description: Closed theorem version of bound-variable hypothesis builder hbn 2295. (Contributed by NM, 10-May-1993.) (Proof shortened by Wolf Lammen, 14-Oct-2021.) | 
| Ref | Expression | 
|---|---|
| hbnt | ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nf5-1 2145 | . . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑) | |
| 2 | 1 | nfnd 1858 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥 ¬ 𝜑) | 
| 3 | 2 | nf5rd 2196 | 1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∀wal 1538 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-or 849 df-ex 1780 df-nf 1784 | 
| This theorem is referenced by: hbn 2295 hbnd 2296 bj-hbext 36711 | 
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