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Mirrors > Home > MPE Home > Th. List > nbn | Structured version Visualization version GIF version |
Description: The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by NM, 21-Jun-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 372 | . . 3 ⊢ (¬ 𝜑 → ((𝜓 ↔ 𝜑) ↔ ¬ 𝜓)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ((𝜓 ↔ 𝜑) ↔ ¬ 𝜓) |
4 | 3 | bicomi 223 | 1 ⊢ (¬ 𝜓 ↔ (𝜓 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 |
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 206 |
This theorem is referenced by: nbn3 374 nbfal 1554 eq0f 4276 eq0ALT 4280 disj 4383 disjOLD 4384 axnulALT 5228 dm0rn0 5829 reldm0 5832 isarchi 31423 |
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