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Mirrors > Home > MPE Home > Th. List > ecase2d | Structured version Visualization version GIF version |
Description: Deduction for elimination by cases. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Dec-2012.) |
Ref | Expression |
---|---|
ecase2d.1 | ⊢ (𝜑 → 𝜓) |
ecase2d.2 | ⊢ (𝜑 → ¬ (𝜓 ∧ 𝜒)) |
ecase2d.3 | ⊢ (𝜑 → ¬ (𝜓 ∧ 𝜃)) |
ecase2d.4 | ⊢ (𝜑 → (𝜏 ∨ (𝜒 ∨ 𝜃))) |
Ref | Expression |
---|---|
ecase2d | ⊢ (𝜑 → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idd 24 | . 2 ⊢ (𝜑 → (𝜏 → 𝜏)) | |
2 | ecase2d.1 | . . . 4 ⊢ (𝜑 → 𝜓) | |
3 | ecase2d.2 | . . . . 5 ⊢ (𝜑 → ¬ (𝜓 ∧ 𝜒)) | |
4 | 3 | pm2.21d 121 | . . . 4 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜏)) |
5 | 2, 4 | mpand 695 | . . 3 ⊢ (𝜑 → (𝜒 → 𝜏)) |
6 | ecase2d.3 | . . . . 5 ⊢ (𝜑 → ¬ (𝜓 ∧ 𝜃)) | |
7 | 6 | pm2.21d 121 | . . . 4 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → 𝜏)) |
8 | 2, 7 | mpand 695 | . . 3 ⊢ (𝜑 → (𝜃 → 𝜏)) |
9 | 5, 8 | jaod 857 | . 2 ⊢ (𝜑 → ((𝜒 ∨ 𝜃) → 𝜏)) |
10 | ecase2d.4 | . 2 ⊢ (𝜑 → (𝜏 ∨ (𝜒 ∨ 𝜃))) | |
11 | 1, 9, 10 | mpjaod 858 | 1 ⊢ (𝜑 → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 ∨ wo 845 |
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 210 df-an 401 df-or 846 |
This theorem is referenced by: (None) |
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