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Mirrors > Home > MPE Home > Th. List > 3imp3i2an | Structured version Visualization version GIF version |
Description: An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.) |
Ref | Expression |
---|---|
3imp3i2an.1 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
3imp3i2an.2 | ⊢ ((𝜑 ∧ 𝜒) → 𝜏) |
3imp3i2an.3 | ⊢ ((𝜃 ∧ 𝜏) → 𝜂) |
Ref | Expression |
---|---|
3imp3i2an | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3imp3i2an.2 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝜒) → 𝜏) | |
2 | 3imp3i2an.1 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | |
3 | 2 | 3exp 1256 | . . . . . . . . . 10 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) |
4 | 3imp3i2an.3 | . . . . . . . . . . 11 ⊢ ((𝜃 ∧ 𝜏) → 𝜂) | |
5 | 4 | ex 449 | . . . . . . . . . 10 ⊢ (𝜃 → (𝜏 → 𝜂)) |
6 | 3, 5 | syl8 74 | . . . . . . . . 9 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 → 𝜂)))) |
7 | 6 | com4r 92 | . . . . . . . 8 ⊢ (𝜏 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))) |
8 | 1, 7 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜒) → (𝜑 → (𝜓 → (𝜒 → 𝜂)))) |
9 | 8 | ex 449 | . . . . . 6 ⊢ (𝜑 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))) |
10 | 9 | pm2.43b 53 | . . . . 5 ⊢ (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))) |
11 | 10 | com4r 92 | . . . 4 ⊢ (𝜒 → (𝜒 → (𝜑 → (𝜓 → 𝜂)))) |
12 | 11 | pm2.43i 50 | . . 3 ⊢ (𝜒 → (𝜑 → (𝜓 → 𝜂))) |
13 | 12 | com3l 87 | . 2 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂))) |
14 | 13 | 3imp 1249 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 |
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 196 df-an 385 df-3an 1033 |
This theorem is referenced by: upgr2pthnlp 40938 av-frgrareg 41548 |
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