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Mirrors > Home > MPE Home > Th. List > 3ianor | Structured version Visualization version GIF version |
Description: Negated triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Ref | Expression |
---|---|
3ianor | ⊢ (¬ (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anor 1054 | . . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒)) | |
2 | 1 | con2bii 347 | . 2 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒) ↔ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒)) |
3 | 2 | bicomi 214 | 1 ⊢ (¬ (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∨ w3o 1036 ∧ w3a 1037 |
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 df-or 385 df-an 386 df-3or 1038 df-3an 1039 |
This theorem is referenced by: tppreqb 4336 funtpgOLD 5943 fr3nr 6979 bropopvvv 7255 prinfzo0 12506 elfznelfzo 12573 ssnn0fi 12784 hashtpg 13267 lcmfunsnlem2lem2 15352 prm23ge5 15520 lpni 27332 xrdifh 29542 dvasin 33496 limcicciooub 39869 2zrngnring 41952 |
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