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Mathbox for Scott Fenton |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 3orit | Structured version Visualization version GIF version |
Description: Closed form of 3ori 1380. (Contributed by Scott Fenton, 20-Apr-2011.) |
Ref | Expression |
---|---|
3orit | ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3or 1032 | . 2 ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒)) | |
2 | df-or 384 | . 2 ⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (¬ (𝜑 ∨ 𝜓) → 𝜒)) | |
3 | ioran 510 | . . 3 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓)) | |
4 | 3 | imbi1i 338 | . 2 ⊢ ((¬ (𝜑 ∨ 𝜓) → 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒)) |
5 | 1, 2, 4 | 3bitri 285 | 1 ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∨ w3o 1030 |
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 196 df-or 384 df-an 385 df-3or 1032 |
This theorem is referenced by: (None) |
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