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Mirrors > Home > MPE Home > Th. List > ecase23d | Structured version Visualization version GIF version |
Description: Deduction for elimination by cases. (Contributed by NM, 22-Apr-1994.) |
Ref | Expression |
---|---|
ecase23d.1 | ⊢ (𝜑 → ¬ 𝜒) |
ecase23d.2 | ⊢ (𝜑 → ¬ 𝜃) |
ecase23d.3 | ⊢ (𝜑 → (𝜓 ∨ 𝜒 ∨ 𝜃)) |
Ref | Expression |
---|---|
ecase23d | ⊢ (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecase23d.1 | . . 3 ⊢ (𝜑 → ¬ 𝜒) | |
2 | ecase23d.2 | . . 3 ⊢ (𝜑 → ¬ 𝜃) | |
3 | ioran 981 | . . 3 ⊢ (¬ (𝜒 ∨ 𝜃) ↔ (¬ 𝜒 ∧ ¬ 𝜃)) | |
4 | 1, 2, 3 | sylanbrc 586 | . 2 ⊢ (𝜑 → ¬ (𝜒 ∨ 𝜃)) |
5 | ecase23d.3 | . . . 4 ⊢ (𝜑 → (𝜓 ∨ 𝜒 ∨ 𝜃)) | |
6 | 3orass 1087 | . . . 4 ⊢ ((𝜓 ∨ 𝜒 ∨ 𝜃) ↔ (𝜓 ∨ (𝜒 ∨ 𝜃))) | |
7 | 5, 6 | sylib 221 | . . 3 ⊢ (𝜑 → (𝜓 ∨ (𝜒 ∨ 𝜃))) |
8 | 7 | ord 861 | . 2 ⊢ (𝜑 → (¬ 𝜓 → (𝜒 ∨ 𝜃))) |
9 | 4, 8 | mt3d 150 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 844 ∨ w3o 1083 |
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 400 df-or 845 df-3or 1085 |
This theorem is referenced by: tz7.7 6185 wfrlem10 7947 archiabllem2b 30875 nolt02o 33312 noresle 33313 nosupbnd1lem6 33326 nosupbnd2lem1 33328 |
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