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| Mirrors > Home > MPE Home > Th. List > imp | Structured version Visualization version GIF version | ||
| Description: Importation inference. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.) |
| Ref | Expression |
|---|---|
| imp.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
| Ref | Expression |
|---|---|
| imp | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-an 396 | . 2 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓)) | |
| 2 | imp.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
| 3 | 2 | impi 164 | . 2 ⊢ (¬ (𝜑 → ¬ 𝜓) → 𝜒) |
| 4 | 1, 3 | sylbi 217 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
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