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Mirrors > Home > MPE Home > Th. List > Mathboxes > syl3an12 | Structured version Visualization version GIF version |
Description: A double syllogism inference. (Contributed by SN, 15-Sep-2024.) |
Ref | Expression |
---|---|
syl3an12.1 | ⊢ (𝜑 → 𝜓) |
syl3an12.2 | ⊢ (𝜒 → 𝜃) |
syl3an12.s | ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜏) → 𝜂) |
Ref | Expression |
---|---|
syl3an12 | ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl3an12.1 | . 2 ⊢ (𝜑 → 𝜓) | |
2 | syl3an12.2 | . 2 ⊢ (𝜒 → 𝜃) | |
3 | id 22 | . 2 ⊢ (𝜏 → 𝜏) | |
4 | syl3an12.s | . 2 ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜏) → 𝜂) | |
5 | 1, 2, 3, 4 | syl3an 1158 | 1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 |
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 206 df-an 396 df-3an 1087 |
This theorem is referenced by: dvdsexpb 40263 |
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