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Mirrors > Home > MPE Home > Th. List > nbn2 | Structured version Visualization version GIF version |
Description: The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by Juha Arpiainen, 19-Jan-2006.) (Proof shortened by Wolf Lammen, 28-Jan-2013.) |
Ref | Expression |
---|---|
nbn2 | ⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (𝜑 ↔ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.501 356 | . 2 ⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (¬ 𝜑 ↔ ¬ 𝜓))) | |
2 | notbi 309 | . 2 ⊢ ((𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓)) | |
3 | 1, 2 | syl6bbr 278 | 1 ⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (𝜑 ↔ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 |
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 197 |
This theorem is referenced by: bibif 361 pm5.21im 364 pm5.18 371 biass 374 sadadd2lem2 15172 isclo 20891 |
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