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Mirrors > Home > ILE Home > Th. List > nfr | GIF version |
Description: Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) |
Ref | Expression |
---|---|
nfr | ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nf 1350 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑)) | |
2 | sp 1401 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (𝜑 → ∀𝑥𝜑)) | |
3 | 1, 2 | sylbi 114 | 1 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1241 Ⅎwnf 1349 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-4 1400 |
This theorem depends on definitions: df-bi 110 df-nf 1350 |
This theorem is referenced by: nfri 1412 nfrd 1413 nfimd 1477 19.23t 1567 equs5or 1711 sbequi 1720 sbft 1728 sbcomxyyz 1846 rgen2a 2375 |
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