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Mirrors > Home > NFE Home > Th. List > nbn | GIF version |
Description: The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2013.) |
Ref | Expression |
---|---|
nbn.1 | ⊢ ¬ φ |
Ref | Expression |
---|---|
nbn | ⊢ (¬ ψ ↔ (ψ ↔ φ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nbn.1 | . . 3 ⊢ ¬ φ | |
2 | bibif 335 | . . 3 ⊢ (¬ φ → ((ψ ↔ φ) ↔ ¬ ψ)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ((ψ ↔ φ) ↔ ¬ ψ) |
4 | 3 | bicomi 193 | 1 ⊢ (¬ ψ ↔ (ψ ↔ φ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 177 |
This theorem is referenced by: nbn3 337 nbfal 1325 n0f 3558 dm0rn0 4921 dmeq0 4922 |
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