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| Mirrors > Home > MPE Home > Th. List > condan | Structured version Visualization version GIF version | ||
| Description: Proof by contradiction. (Contributed by NM, 9-Feb-2006.) (Proof shortened by Wolf Lammen, 19-Jun-2014.) |
| Ref | Expression |
|---|---|
| condan.1 | ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜒) |
| condan.2 | ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ 𝜒) |
| Ref | Expression |
|---|---|
| condan | ⊢ (𝜑 → 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | condan.1 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜒) | |
| 2 | condan.2 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ 𝜒) | |
| 3 | 1, 2 | pm2.65da 817 | . 2 ⊢ (𝜑 → ¬ ¬ 𝜓) |
| 4 | 3 | notnotrd 133 | 1 ⊢ (𝜑 → 𝜓) |
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