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Mirrors > Home > MPE Home > Th. List > 3oran | Structured version Visualization version GIF version |
Description: Triple disjunction in terms of triple conjunction. (Contributed by NM, 8-Oct-2012.) |
Ref | Expression |
---|---|
3oran | ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3ioran 1103 | . . 3 ⊢ (¬ (𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒)) | |
2 | 1 | con1bii 356 | . 2 ⊢ (¬ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒) ↔ (𝜑 ∨ 𝜓 ∨ 𝜒)) |
3 | 2 | bicomi 223 | 1 ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∨ 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 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 |
This theorem is referenced by: nolt02o 27568 nogt01o 27569 nosupbnd1lem6 27586 noinfbnd1lem6 27601 dalawlem10 39254 |
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