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Mirrors > Home > ILE Home > Th. List > syl2ani | GIF version |
Description: A syllogism inference. (Contributed by NM, 3-Aug-1999.) |
Ref | Expression |
---|---|
syl2ani.1 | ⊢ (𝜑 → 𝜒) |
syl2ani.2 | ⊢ (𝜂 → 𝜃) |
syl2ani.3 | ⊢ (𝜓 → ((𝜒 ∧ 𝜃) → 𝜏)) |
Ref | Expression |
---|---|
syl2ani | ⊢ (𝜓 → ((𝜑 ∧ 𝜂) → 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl2ani.1 | . 2 ⊢ (𝜑 → 𝜒) | |
2 | syl2ani.2 | . . 3 ⊢ (𝜂 → 𝜃) | |
3 | syl2ani.3 | . . 3 ⊢ (𝜓 → ((𝜒 ∧ 𝜃) → 𝜏)) | |
4 | 2, 3 | sylan2i 405 | . 2 ⊢ (𝜓 → ((𝜒 ∧ 𝜂) → 𝜏)) |
5 | 1, 4 | sylani 404 | 1 ⊢ (𝜓 → ((𝜑 ∧ 𝜂) → 𝜏)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: disjxp1 6215 mapen 6824 mgmidmo 12626 |
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