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| Mirrors > Home > MPE Home > Th. List > pm2.61dane | Structured version Visualization version GIF version | ||
| Description: Deduction eliminating an inequality in an antecedent. (Contributed by NM, 30-Nov-2011.) |
| Ref | Expression |
|---|---|
| pm2.61dane.1 | ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝜓) |
| pm2.61dane.2 | ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝜓) |
| Ref | Expression |
|---|---|
| pm2.61dane | ⊢ (𝜑 → 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pm2.61dane.1 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝜓) | |
| 2 | 1 | ex 412 | . 2 ⊢ (𝜑 → (𝐴 = 𝐵 → 𝜓)) |
| 3 | pm2.61dane.2 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝜓) | |
| 4 | 3 | ex 412 | . 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 → 𝜓)) |
| 5 | 2, 4 | pm2.61dne 3028 | 1 ⊢ (𝜑 → 𝜓) |
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