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Mirrors > Home > MPE Home > Th. List > hbnt | Structured version Visualization version GIF version |
Description: Closed theorem version of bound-variable hypothesis builder hbn 2291. (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 2140 | . . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑) | |
2 | 1 | nfnd 1860 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥 ¬ 𝜑) |
3 | 2 | nf5rd 2188 | 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 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-10 2136 ax-12 2170 |
This theorem depends on definitions: df-bi 206 df-or 845 df-ex 1781 df-nf 1785 |
This theorem is referenced by: hbn 2291 hbnd 2292 bj-hbext 34988 |
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