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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-nfnbi | Structured version Visualization version GIF version |
Description: Being free does not depend on an outside negation in an expression. This theorem is slightly more general than nfn 1768 or nfnd 1769. (Contributed by Wolf Lammen, 5-May-2018.) |
Ref | Expression |
---|---|
wl-nfnbi | ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfnt 1767 | . 2 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑) | |
2 | notnotb 303 | . . 3 ⊢ (𝜑 ↔ ¬ ¬ 𝜑) | |
3 | nfnt 1767 | . . 3 ⊢ (Ⅎ𝑥 ¬ 𝜑 → Ⅎ𝑥 ¬ ¬ 𝜑) | |
4 | 2, 3 | nfxfrd 1772 | . 2 ⊢ (Ⅎ𝑥 ¬ 𝜑 → Ⅎ𝑥𝜑) |
5 | 1, 4 | impbii 198 | 1 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 Ⅎwnf 1699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 |
This theorem depends on definitions: df-bi 196 df-or 384 df-ex 1696 df-nf 1701 |
This theorem is referenced by: wl-sb8et 32513 |
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