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Mirrors > Home > MPE Home > Th. List > pm2.61iii | Structured version Visualization version GIF version |
Description: Inference eliminating three antecedents. (Contributed by NM, 2-Jan-2002.) (Proof shortened by Wolf Lammen, 22-Sep-2013.) |
Ref | Expression |
---|---|
pm2.61iii.1 | ⊢ (¬ 𝜑 → (¬ 𝜓 → (¬ 𝜒 → 𝜃))) |
pm2.61iii.2 | ⊢ (𝜑 → 𝜃) |
pm2.61iii.3 | ⊢ (𝜓 → 𝜃) |
pm2.61iii.4 | ⊢ (𝜒 → 𝜃) |
Ref | Expression |
---|---|
pm2.61iii | ⊢ 𝜃 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.61iii.4 | . 2 ⊢ (𝜒 → 𝜃) | |
2 | pm2.61iii.1 | . . 3 ⊢ (¬ 𝜑 → (¬ 𝜓 → (¬ 𝜒 → 𝜃))) | |
3 | pm2.61iii.2 | . . . 4 ⊢ (𝜑 → 𝜃) | |
4 | 3 | a1d 25 | . . 3 ⊢ (𝜑 → (¬ 𝜒 → 𝜃)) |
5 | pm2.61iii.3 | . . . 4 ⊢ (𝜓 → 𝜃) | |
6 | 5 | a1d 25 | . . 3 ⊢ (𝜓 → (¬ 𝜒 → 𝜃)) |
7 | 2, 4, 6 | pm2.61ii 186 | . 2 ⊢ (¬ 𝜒 → 𝜃) |
8 | 1, 7 | pm2.61i 185 | 1 ⊢ 𝜃 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem is referenced by: axrepnd 10208 axacndlem4 10224 axacndlem5 10225 axacnd 10226 |
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