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Mirrors > Home > MPE Home > Th. List > syl2ani | Structured version Visualization version 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 604 | . 2 ⊢ (𝜓 → ((𝜒 ∧ 𝜂) → 𝜏)) |
5 | 1, 4 | sylani 602 | 1 ⊢ (𝜓 → ((𝜑 ∧ 𝜂) → 𝜏)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 |
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 395 |
This theorem is referenced by: 2mo 2639 frxp 8135 poxp2 8152 mapen 9170 rex2dom 9275 fin1a2lem9 10437 coprmproddvdslem 16638 psss 18577 mgmidmo 18625 aannenlem1 26281 funtransport 35632 cgrxfr 35656 btwnxfr 35657 bj-cbv3tb 36269 |
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