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| Mirrors > Home > MPE Home > Th. List > df-3an | Structured version Visualization version GIF version | ||
| Description: Define conjunction ('and') of three wff's. Definition *4.34 of [WhiteheadRussell] p. 118. This abbreviation reduces the number of parentheses and emphasizes that the order of bracketing is not important by virtue of the associative law anass 468. (Contributed by NM, 8-Apr-1994.) |
| Ref | Expression |
|---|---|
| df-3an | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | wph | . . 3 wff 𝜑 | |
| 2 | wps | . . 3 wff 𝜓 | |
| 3 | wch | . . 3 wff 𝜒 | |
| 4 | 1, 2, 3 | w3a 1087 | . 2 wff (𝜑 ∧ 𝜓 ∧ 𝜒) |
| 5 | 1, 2 | wa 395 | . . 3 wff (𝜑 ∧ 𝜓) |
| 6 | 5, 3 | wa 395 | . 2 wff ((𝜑 ∧ 𝜓) ∧ 𝜒) |
| 7 | 4, 6 | wb 206 | 1 wff ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) |
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