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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 1448 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑)) | |
2 | sp 1498 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (𝜑 → ∀𝑥𝜑)) | |
3 | 1, 2 | sylbi 120 | 1 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1340 Ⅎwnf 1447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-4 1497 |
This theorem depends on definitions: df-bi 116 df-nf 1448 |
This theorem is referenced by: nfri 1506 nfrd 1507 nfimd 1572 19.23t 1664 equs5or 1817 sbequi 1826 sbft 1835 sbcomxyyz 1959 rgen2a 2518 |
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