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Mirrors > Home > MPE Home > Th. List > mp3an1i | Structured version Visualization version GIF version |
Description: An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.) |
Ref | Expression |
---|---|
mp3an1i.1 | ⊢ 𝜓 |
mp3an1i.2 | ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)) |
Ref | Expression |
---|---|
mp3an1i | ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mp3an1i.1 | . . 3 ⊢ 𝜓 | |
2 | mp3an1i.2 | . . . 4 ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)) | |
3 | 2 | com12 32 | . . 3 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → (𝜑 → 𝜏)) |
4 | 1, 3 | mp3an1 1403 | . 2 ⊢ ((𝜒 ∧ 𝜃) → (𝜑 → 𝜏)) |
5 | 4 | com12 32 | 1 ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → 𝜏)) |
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: (None) |
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