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Mirrors > Home > MPE Home > Th. List > 3anor | Structured version Visualization version GIF version |
Description: Triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Wolf Lammen, 8-Apr-2022.) |
Ref | Expression |
---|---|
3anor | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3ianor 1104 | . . 3 ⊢ (¬ (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒)) | |
2 | 1 | con1bii 360 | . 2 ⊢ (¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒) ↔ (𝜑 ∧ 𝜓 ∧ 𝜒)) |
3 | 2 | bicomi 227 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∨ w3o 1083 ∧ w3a 1084 |
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 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 |
This theorem is referenced by: ne3anior 3080 swrdnd0 14010 |
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